Problem Solving Strategies: Applications of the “Choose 2” method

1. Example: How many diagonals can be drawn for a polygon with “n” sides?

Method I:

The number of diagonals in a polygon = n(n-3)/2, where n is the number of polygon sides.

For a convex n-sided polygon, there are n vertexes, and from each vertex you can draw n-3 diagonals, so the total number of diagonals that can be drawn is n (n-3).

However, this would mean that each diagonal would be drawn twice, (to and from each vertex), so the expression must be divided by 2.

Method II:

nC2 (choose 2) - n sides = n(n-1)/ 2 – n sides


2. Example: There are n people at a party, each person shakes hands with the every other person once. How many handshakes?

Method I:
nC2 in this case (10 x 9) /2 =45
__ __ First slot you have 10 persons to choose from, second slot 9 persons. Since A shakes hands with B is the same as B shakes hands with A, so you divide the number by 2 and get the answer.

Method II:
Sum of the first consecutive Natural numbers: n (n+1) /2
The first person shakes hands with 9 other person; the second person shakes hands with 8 other person, etc…
So 9 + 8 + 7 + …= (9 x 10)/ 2 = 45

3. Example: N dots evenly spaced on a circle. How many chords can you make using those dots?

Methods: This is very similar to hand-shaking questions.

I: nC2

II: Sum of the first consecutive (N-1) Natural numbers

Word problems: Answers below.

#1 : A convex polygon with n sides has 20 diagonals. How many diagonals does an (n+1)-sided convex polygon have?

#2: A polygon has n sides and n diagonals. What is n? 

#3: How many diagonals does a decagon have?

#4: How many diagonals does a dodecagon have? 

#5: How many line segments have both their endpoints located at the vertexes of a given cube?

#6: There are 8 points on a circle, how many lines can you make? How many triangles can you make?

#7: If each of the interior angles of two regular polygons adds up to 255 degrees and their diagonals add up to 29, what is the sum of their sides?

Answers: 

# 1- 27    #2 -5     #3-35      #4-54 [12C2 -12]     #5 28 [There are 8 vertices, so 8C2]



#6  8C2 = 28 for lines and 8C3 = 56 triangles   

#7: 6[hexagon] + 8[octagon] = 14 sides

Unit digit, Tenth digit and Digit Sum

Word problems on unit digit, tenth digit or digit sum.

#1: How many digits are there in the positive integers 1 to 99 inclusive? 

Solution I:  From 1 to 9, there are 9 digits.
From 10 to 99, there are 99 - 10 + 1 or 99 - 9 = 90 two digit numbers. 90 x 2 = 180
Add them up and the answer is 189.

Solution II:  ___ There are 9  one digit numbers (from 1 to 9).
___ ___ There are 9 * 10 = 90 two digit numbers (You can't use "0" on the tenth digit but you
can use "0" on the unit digit.) 90 * 2 + 9 = 189

# 2: A book has 145 pages. How many digits are there if you start counting from page 1?

There are 189 digits from page 1 to 99. (See #1, solution I)
From 100 to 145, there are 145 - 100 + 1 or 145 - 99 = 46 three digit numbers.
189 + 46*3 = 327 digits.

#3: "A book has N pages, number the usual way, from 1 to N. The total number of digits in the page number is 930. How many pages does the book have"?  Similar to one Google interview question.

Read the questions and others here from the Wall Street Journal.

Solution I: 

930 - 189 (digits of the first 99 pages) =741
741 divided by 3 = 247. Careful since you are counting the three digit numbers from 100 if the book has N
pages N - 100 + 1 or N - 99 = 247. N = 346 pages.

Solution II: 

930 - 189 (total digits needed for the first 99 pages) = 741
741/3 = 247 (how far the three digit page numbers go).
247 + 99 = 346 pages

#4: If you write consecutive numbers starting with 1, what is the 50th digit you write? 

Solution I:
50 - 9 = 41, 9 being the first 9 digits you need to use for the first 9 pages.

Now it's 2 digit. 41/2 = 20.1 , which means you will be able to write 20 two digit numbers + the first digit of the next two digit numbers.

10 to 29 is the first 20 two digit numbers so the next digit 3 is the answer. (first digit of the two digit number 30.)

Solution II: (50 - 9 ) / 2 = 20. 5 ; 20.5 + 9 = 29.5, so 29 pages + the first digit of the next two digit numbers, which is 3, the answer.


#5: What is the sum if you add up all the digits from 1 to 100 inclusive?

00  10  20  30  40  50  60  70  80  90

01  11  21  31  41  51  61  71  81  91

02  12  22  32  42  52  62  72  82  92

03  13  23  33  43  53  63  73  83  93

04  14  24  34  44  54  64  74  84  94

05  15  25  35  45  55  65  75  85  95

06  16  26  36  46  56  66  76  86  96

07  17  27  37  47  57  67  77  87  97

08  18  28  38  48  58  68  78  88  98

09  19  29  39  49  59  69  79  89  99

Solution I:
Do you see the pattern?  From 00 to 99 if you just look at the unit digits.
There are 10 sets of ( 1+ 2 + 3 ... + 9) , which gives you the sum of 10 * 45 = 450
How about the tenth digits? There are another 10 sets of (1 + 2 + 3 + ...9) so another 450
Add them up and you have 450 * 2 = 900 digits from 1 to 99 inclusive.
900 + 1 ( for the "1" in the extra number 100) = 901 

Solution II:
If you add the digits on each column, you have an arithmetic sequence, which is
45 + 55 + 65 ... + 135  To find the sum, you use average * the terms (how many numbers)
\(\dfrac {45+135} {2} * \left( \dfrac {135-45} {10}+1\right)\) =900
900 + 1 = 901

Solution III :
2*45*10¹ + 1 = 901


Problems to practice: Answers below.

#1: A book has 213 pages, how many digits are there?

#2: A book has 1012 pages, how many digits are there?

#3: If you write down all the digits starting with 1 and in the end there are a: 100, b: 501 and c: 1196 digits, what is the last digit you write down for each question?

#4: What is the sum of all the digits counting from 1 to 123? 

Answers: 

#1: 531 digits. 

#2: 2941 digits.

#3: a. 5, b. 3, c. 3

#4: 1038 

Ratio, Proportions -- Beginning Problem Soving (SAT level)

Questions:
#1: At a county fair, if adults to kids ratio is 2 to 3 and there are 250 people at the fair. How many adults and how many kids? 

Solution I :  Adults to Kids = 2 : 3 (given). Let there be 2x adults and 3x kids (only when two numbers have the same multiples can you simplify the two number to relatively prime.
 2x + 3x = 250;  5x = 250; x = 50 so there are 2x or 2 * 50 = 100 adults and 3x or 3 * 50 = 150 kids.

Solution II:  From the given information, you know that every time when there are 5 parts (2 + 3 = 5), there
will be 2 parts for A, or   of the total, and 3 parts for B, or of the total.
  * 250 = 100 adults and * 250 = 150 kids

#2: In a mixture of peanuts and cashews, the ratio by weight of peanuts to cashews is 5 to 2. How many pounds of cashews will there be in 4 pounds of this mixture? (an actual SAT question)

Solution:
You can use either the first or second method (see above two solutions) but the second solution is much faster. 
* 4 = pounds

#3: Continue with question #1: How many more kids than adults go to the fair? 

Solution I: 
Use the method on #1 and then find the difference.
150 - 100 = 50 more kids.
  
Solution II: 
Using the method for #1: solution II, you know    of the all the people go to the fair are adults ,
and of the total people go to the fair are kids.
(  -   ) * 250 = 50 more kids.

#3: If the girls to boys ratio at an elementary school is 3 to 4 and there are 123 girls, how many boys are there at that elementary school? 
Solution:
This one is easy, you set up the equation. Just make sure the numbers line up nicely.
Let there be "x" boys. = . Cross multiply to get x. Or since 123 is 41 * 3;
4 * 41 = 164 boys

#4 : If the girls to boys ratio at an elementary school is 2 to 5 and there are 78 more boys than girls, how many girls are there at the elementary school? 

Solution I: 
Again, you can use the algebra and let there be 2x girls and 5x boys. 
According to the given, 5x - 2x = 3x = 78 so x = 26
Plug in and you get there are 2*26 = 52 girls and 5*26 = 130 boys.

Solution II: 
If you keep expanding the ratio using the same multiples, 2 : 5 = 4 : 10 : 6 : 15...
Do you see that the difference of boys and girls are always multiples of 3 ( 5 - 2  = 3).
  = 26 so there are 2 * 26 or 52 girls

Mental Math Practices on Ratio, Percents and Proportion

Mental Math Practices: All questions can be solved in your head reasonably fast (less than 5 minutes) if you understand the concepts well.

Check out Mathcounts here, the best competition math program for middle school students.
Download this year's Mathcounts handbook here.

1. What is 12.5% of 128?

2. After 20% discount, you paid $144? What is the original price?

3. To make 20% profit, you charge your customer 144 dollars. What is the wholesale price?

4. You lost 20% in the stock market. What percentage increase will you break even?

5. A : B = 2 : 5 and A is 75 more than B, what is B?

6. A : B = 2 : 3 and A + B = 14, what is A and what is B?

7. A : B = 3 : 5 and B : C = 2 to 9, what is A : B?

8. Two triangles are similar and the length of their base ratio is 1: 2. The smaller triangle has a base that equals 5 and an area that is 110. What is the area of the larger triangle?

9. One side of the square increases 10% and the other side decreases 20 percent. What is the percentage change? Increase or decrease? By how much?

10. There are 60 students in a gym and 30% are boys. After a few more boys enter the gym, now 40% are boys. How many boys enter the room?

Mathcounts strategies: Some sums

The following sequences are all arithmetic sequences and for any arithmetic sequences, the sum is 

always average times the terms (how many numbers). 

To find the average, you can use 

a. sum divided by how many numbers.
b. average of the first and the last term, the second first and the second last term, or the third first and the third last term, etc...

Sum of the first consecutive natural numbers:

1 + 2 + 3 + 4 + 5 + ... + ( n -1 ) + n = \( \dfrac {n\left( n+1\right) } {2}\) 

Examples : 

#1: 1 + 2 + 3 + ... + 100 = \(\dfrac {100\left ( 101\right) } {2}=5050\)

#2: 1 + 2 + 3 + ... + 27 = \( \dfrac {27\left( 28\right) } {2}=378\)

#3: 4 + 8 + 12 + ... + 80 = 4 (1 + 2 + 3 + ... + 20) = \(\dfrac {4\times 20\times 21} {2}=840\)

Sum of the first consecutive natural even numbers: Proof without words 

Have you noticed in this sequence, every number is double the numbers in the first example,

so you don't need to divide by 2.  

\(2+4+6+\ldots +2n=n\left( n+1\right) \)

Examples: 

#1: 2 + 4 + 6 + ... 100 = 50 * 51 = 2550

#2: 2 + 4 + 6 + 8 + 10 + 12 = 6 x 7 = 42 

In this case, you can also find the midpoint, which is 7 and then 7 x 6 = 42

#3: 2 + 4 + 6 + ...420 = 210 * 211 = 44,310

Sum of the first consecutive odd numbers: Proof without words

In this special case, the mean is the same as how many numbers.

So it's easier if you find the mean by averaging the first and the last term and then square the mean.

1 + 3 + 5 + ... + ( 2n -1) = \(n^{2}\)

Examples: 

#1: 1 + 3 + 5 + ... + 39 = \(\left[ \dfrac {\left( 39+1\right) } {2}\right] ^{2}\) = (20^{2}\)

#2: 1 + 3 + 5 + ... + 89 = \(\left[ \dfrac {89+1} {2}\right] ^{2}=45^{2}=2025\)

#3: 1 + 3 + 5 + ... + 221 = \(\left[ \dfrac {221+1} {2}\right] ^{2}= 111^{2}=12321 \)

Applicable problems:

#1: What is the sum of the first 40 consecutive positive integers? 

#2: What is the sum of the first 40 consecutive positive even integers?

#3: What is the sum of the first 40 consecutive positive odd integers? 

#4: How many times does a 12-hour clock strikes in one day if it strikes once on one o'clock, twice on two o'clock, etc...?

#5: At a game show,you win $100 for the first correct answer and $200 for the second correct answer, etc. How much do you win if you answer 8 questions in a row correctly? 

Answer key:

#1: 820

#2: 1640

#3: 1600

#4: 156

#5: 100 + 200 + 300 ... + 800 = 100 (1 + 2 + 3 + ...8) = 100* \(\dfrac {8\times 9} {2}\) =  3600 

Painted Cube Problems: Beginning Level

Please refrain yourself from checking the answers too soon. Use Lego blocks, unit cubes, or Rubik's cubes to help you think. If you are really stuck, check out this link on painted cube problems.

Please comment and help me make my blog more user friendly. Thanks a lot!! 

Painted Cube Problem, competition math, problem solving by Mrs. Lin

Answer Key to the Painted Cube Problem

Prime Factorization : Part II

   Learn basic facts here.











Another fun trick is to find the total number of factors any number has. To do this, take the prime factorization of the number, add 1 to each of the exponents of the prime factor, and multiply these new numbers together.

Example:

#1: 12 = 2² x 3¹
(2+1) x (1+1) = 3 x 2 = 6
Therefore, 12 has 6 factors. {1, 2, 3, 4, 6, 12}

#2: 8= 2³ , (3+1) = 4 Therefore, 8 has 4 factors. {1, 2, 4, 8}

Problems to ponder: answers and solutions below.
1. What is the smallest number with 5 factors?
2. What is the smallest number with 6 factors?
3. What is the smallest number with 12 factors?
4. How many factors does any prime number have?
5. How many even factors does 36 have? (tricky question)
6. How many perfect square factors does 144 have? ( don’t just list them, think how you get the answers)









Answers:
1. 16  Since 5 itself is a prime, the smallest number that has 5 factors would look like this: n⁴  put in the smallest prime number and the answer is  2⁴ = 16 

2. 12  The number 6 can be factor as 6 x 1 or 3 x 2 so you can have either n⁵ or x² * y¹
so either 2⁵ or 2² * 3 so the answer is 12. 

3. 60  12 = 12 x 1 = 6 x 2 = 4 x 3 = 3 x 2 x 2  The last would give you the smallest number which is  
2² * 3¹ * 5¹ = 60

4. 2 Any prime number has two factors, which are 1 and itself. The smallest prime number is 2, which is the oddest prime -- the only even number that is a prime. (Why??)

5. 6  Any even number multiples another integer will give you another even number. To get only even factors, you  need to always leave the smallest even number, which is 2, with all the other factors. 
36 = 2² * 3²
= 2 (2¹ *3²)  There are(1 + 1) ( 2 + 1 ) = 6 even factors

6. 6  The number 144 = 2⁴ * 3²
To get only square factors, you need to keep the smallest square number of all the prime numbers, leave out the others that are not square and find how many factors that new arrangement has. 
( 2²)² * (3²)¹  There are (2 + 1) ( 1 + 1)= 6 square factors.

See what they are on the left.

Special Right Triangles: 30-60-90 and 45-45-90 Degrees Right Triangles

Please give me feedback on the comment.

Thanks a lot!!  Mrs. Lin







These are the two most common right triangles.

For 45-45-90 degrees, the ratio is  1 - 1 - √ 2

For 30-60-90 degrees, the ratio is  1 - √ 3 - 2

Other concepts to remember are that in any triangle a. larger angle corresponds to longer side and b.same 
angles have the same side length.

Level 0 skills check practices: 

Find the missing side length (answers below)

I: 45-45-90 degrees right triangle

a. 3 - ___- ___            b. 5 - ___- ___         c.  √ 2 - ___ - ___      d. tricky : ___ - ___ - √3

e. ___ - ___ - 4           f. ___ - ___ - 6

II. 30-60-90 degrees right triangle

a. 3 - ___- ___            b.  ___- ___  - 4        c.  √ 2 - ___ - ___      d. ___ - ___ - 5

e.  ___ - 6 - ___           f. ___ - 3 - ___

Answer key and some notes:

I: 45-45-90 degrees right triangle

a. 3 - 3- 3 √ 2              b. 5 - 5 - 5 √ 2         c.  √ 2 - √ 2  - 2       d.√ 6 /2 - √ 6 /2 - √3

e. 2√ 2 - 2√ 2 - 4         f. 3√ 2 - 3√ 2  - 6

Notes:
a. Given the side length to 45 degrees, the easiest way to get the 90 degree side length is to time that number by √ 2 .

b. Given the side length to 90 degrees, the easiest way to get the 45 degree side length is to divide that number by 2 and then times √ 2 .

II. 30-60-90 degrees right triangle

a. 3 - 3√3 - 6            b.  2 - 2√3 - 4        c.  √ 2 - √ 6  - 2√ 2       d. 2.5 - 2.5√3 - 5

e.  2√3 - 6 - 4√3         f.  √3 - 3 -  2√3

Notes:
a. Given the side length to 30 degrees, the easiest way to get 90 degrees side length is to times 2 to the 30 degree side length. To get the side length to 60 degrees, times √3 to the side length to 30 degrees.

 b. Given the side length to 90 degrees, divide 90 degree side length by 2 to get the side length to 30 degrees.Times √3 to the side length of 30 degrees to get the side length of 60 degrees.

c. Given the side length to 60  degrees, divide that number by 3 and then times √3  to get the side length of 30 degrees. Times 2 to get the side length of 90 degrees.

How Many Numbers (Terms)? Space, Inclusive and Exclusive Notes

Please write a comment and give me feedback. Thanks a lot!!

Quite a lot of my students have problems figuring out this type of problems so here are the notes.

#1: How many consecutive numbers from 1 to 5 inclusive?

1 _ 2 _ 3 _ 4 _ 5   There are 5 numbers if you just list them and count them out; however, what if
the question is:

#2: How many consecutive numbers from 34 to 200 (SAT type problem)?

Most students would think it's 200 - 34 = 166, but it's not.

Using #1 case, if you do 5 - 1 = 4, you are only getting how many spaces between those consecutive numbers.

Thus for question #2, the correct answer is 200 - 34 + 1 or 200 - 33 = 167

#3: What about how many consecutive numbers from 5 to 100 exclusive?

Inclusive means including the first and the last numbers; exclusive means not including the first and the last numbers, so for this question, you do 100 - 5 - 1 = 94.

Use # 1 case to help you figure out and really understand the concepts involved.

Here are other questions to help you practice the skills.

Word problems:  Answers below.

#1: How many numbers from 45 to 100 inclusive? 

#2: How many numbers from 17 to 127 inclusive? 

#3: How many numbers from 12 to 34 exclusive? 

#4: How many multiples of 9 from 1 to 200 inclusive?

#5: The dimension of the square on the left is 20 feet by 20 feet. If you  place a post every four feet, starting at one corner, how many posts will be placed? 

#6:  The distance from exit 13 to 21 is 216 miles. How many miles is the distance between two exits if all exits are equally spaced? 

#7: How many multiples of 5 from 120 to 218 exclusive? 

#8: Who is right? The teacher or the student?  Try this question.

#9: How many numbers from -12, -11, -10.........56 inclusive? 

What is their sum?

#10: How many numbers are in the list: 17.25, 18.25, 19, 25...111.25? 

Solutions: To excel at Mathcounts state/national, you need to practice all these questions mentally.

#1: 100 - 45 + 1 = 100 - 44 = 56
#2: 127-17 + 1 = 127 - 16 = 111
#3: Exclusive: 34 -12 -1 = 34 - 13 = 21
#4: Multiples of 9 from 1 to 200 starts with 9 and ends in 198.

Solution I: 9 , 18, 27...198  = 9 (1, 2, 3, ...22) The answer is 22.

Solution II:   \(\frac{(198 - 9)}{9} + 1 = 22"\)

#5:  Just observe one side first. Exclude the 4 corners, the other posts
are similar to those exclusive type problems.
There are posts on each side so 4 * 4 + 4 (corner posts)
= 20



 #6: There are 21 - 13 = 8 space so miles.  The answer is 27 miles.
#7:  Multiples of 5 from 120 to 218 start with 120 and end in 215.
Since it's asking exclusive, 120, 125, ...215 = 5(24, 25, ...43)
43 - 24 - 1 = 43 - 25 = 18
#8: You only need two cuts to get 3 pieces so 2 * 10 = 20 minutes. The student is right.
#9: 56 - ( -12) + 1 = 56 + 13 = 69
The sum is from 13 to 56 since up to 12 it got cancelled with the negative equivalent numbers.
Use average * the term you got  . The sum is 1518.
#10:111.25 - 117.25 + 1 = 111.25 - 116.25 = 95

Problem Solving Strategy: Counting Coins

Q: How many ways can you make 25 cents if you can use quarters, dimes, nickels or pennies?
Start with the largest value:

1 Quarter      1 way

2 Dimes, 1 Nickel   You can stop here since it implies 2 ways.
2 Dimes, 0 Nickel ( which implies 5 pennies)

1 Dime, 3 Nickels , which implies 4 ways.
1 Dime, 2 Nickels ( 5 pennies)
1 Dime, 1 Nickel   (10 pennies)
1 Dime, 0 Nickel ( 15 pennies)

0 Dime, 5 N, which implies 6 ways.
: , 4 N  (5 pennies)
: , 3 N  (10 pennies)
: , 2 N  (15 pennies)
: , 1 N  (20 pennies)
: , 0 N (25 pennies)

So altogether 13 ways.

Practice questions: (answers below)

Q 1 : How many ways can you make a. 15cents, b. 20cents, c. 30 cents if you can use quarters, dimes, nickels or pennies?

Q 2 : How many different combination of coins could a person have if she has exactly 21 cents?

Q 3 :Using nickels, dimes, quarters and/or half-dollars, how many ways can you make 75 cents?

Q 4: 20 coins of quarters and nickels add up to 4 dollars. How many nickels are there? 

Q5:  What is the least number of US coins to make changes possible from 1 to  99 cents inclusive? (half dollar is allowed)










Answers: 

#1: a. 6 ways ; b. 9 ways; c 18 ways ;
#2: 9 ways
#3: 22  ways
#4: 5 nickels and 15 quarters   
#5: 9 coins (1 half dollar, 1 quarter, 2 dimes, 1 nickel and 4 pennies)

Prime Numbers: Mathcounts Beginning Level

This year's Mathcounts' handbook can be downloaded free here .

Please take a look at what that program is all about. It's team work, problem solving, fun, friendship building and lots and lots more. The majority of students we met at the Mathcounts Nationals all went to the most selected colleges and are thriving there.

Useful Definitions:

Prime – a number which cannot be divided by any numbers other than 1 and itself.

Factors – all whole numbers which can evenly divide a given number

Factoring – the breakdown of any number into its prime components

Greatest Common Factor (GCF)– the greatest number which is a factor of two or more given numbers

Least Common Multiple/Denominator (LCM)– the smallest number which is a multiple of two or more given numbers

Relatively Prime – two numbers with a GCF of 1

A prime, as stated in the list of useful definitions, is a number which cannot be divided by any numbers other than 1 and itself. The smallest prime is 2. [Or, as some people claim, the oddest prime.]

Whole numbers which are not primes are called composite. The smallest composite number is 4.

1 is the exception: it is considered to be neither a prime nor a composite number.

Given a chart of the whole numbers 2-100, the primes can be easily recognized:

The easiest thing to do is to the look at the smallest primes – namely, 2, 3, 5, 7 – and cross all multiples of them from the chart. 

The numbers most commonly mistaken for primes are 51, 57 and 91. The first two (51 and 57) as can be shown by adding up the digits, is divisible by 3, while 91 is equal to 7x13. 

To decide whether or not a number is a prime, take its square root and try dividing the original number by all primes less than the square root. If it is not divisible by any of them, the number is a prime. 

Questions: (beginning level) 

#1: List all the two digit prime numbers that end in a. unit digit 1. b. unit digit 3. c. unit digit 7. d. unit digit 9.

#2: What is the smallest prime number that is the sum of two prime square numbers? 

#3: An emirp (prime spelled backwards) is a prime that gives you a different prime when its digits are reversed. What is the smallest emirp? What are all of the emirps between 1 and 100 inclusive?

#4: The number "p" has three distinct prime factors. How many factors does the number "p" have?

#5: What is the smallest number that has 5 factors? 7 factors? 11 factors? Any pattern?

#6: What is the smallest number that has 6 factors? 10 factors? 12 factors? 20 factors?

#7: How many positive factors does the number 24 have?

#8: Find the sum of all the positive factors of 24?

#9: The GCF (greatest common factor) of x and 21 is "3". If x is smaller than 200, how many possible x are there?

#10: How many even factors does the number 180 have?

Solutions:

#1: See the prime number chart.

#2:  2² + 3² = 13

#3:  13 ( 31 is the reverse prime). The other emirps below 100 are 17, 31, 37, 71, 73, 79, and 97.

#4:  To fine how many factors a number has, you prime factorize that number and add one to each of the
exponents of those prime and multiply them together.
Let x, y, z be the three prime numbers that make up the number p.
p = x * y * z  (1 + 1) (1 + 1) (1 + 1) = 8 -- each prime number has exponent 1.
The 8 factors of p are 1, x, y, z, xy, xz, yz, xyz (or p)

#5: The smallest number that has 5 factors is 2⁴ or 16. The factors are 1, 2, 4, 8, 16.
Since the exponent is 4, there are (4 + 1) = 5 factors.
The smallest number that has 7 factors is 2⁶ or 64.
The smallest number that has 11 factors is 2¹⁰ = 1024
5, 7, 11 are all prime numbers. 

#6: This one is harder than the previous question. 6 =1*6 or 3 * 2
2⁵ = 32  ;   2² * 3 = 12  (both numbers have 6 factors but the latter is much smaller).
The answer is 12.

10 = 1 x 10 = 5 x 2  ;  2⁴ * 3 = 48

12 = 1 x 12 = 6 x 2 = 4 x 3 = 3 x 2 x 2
2²* 3 * 5 = 60

20 = 1 x 20 = 2 x 10 = 5 x 2 x 2
2⁴* 3 * 5 = 240

#7: 24 = 3¹*2³ (1 + 1) (3+1) =8; Those factors are:

#8: Continue to the previous number 24, the easiest way to find the sum is to use the
following method:
(1 + 2 + 4 + 8) (1 + 3) = 60

#9: 

#10:  180 = 2²* 3 ²* 5
To get just the even factors, you need to keep a 2 to guarantee the factors stay even.
2 ( 2 * 3²* 5)  There are (1+1)(2+1)(1+1) = 12 factors.