Mental Math Tricks I

Multiples of 11:

11 x 14 = 1 ___ 4 , the middle number is (1 + 4) so the answer is 154

Let's try a few calculations mentally.

11 x 33 = 3 __ 3, the middle number is ( 3 + 3 ) so the answer is 363

11 x 52 = 5 __ 2, the middle number is ( 5+2 ) so the answer is 572

11 x 27 = 2 __ 7 , the middle number is ( 2 + 7 ) so the answer is 297

What about if the sum of the middle number is > 9? Then you carry over the "1" to the digit on its left
as you are doing the addition.

Examples:

11 x 67 = 6 __ 7, the middle number is (6 + 7) = 13 so the answer is 737

11 x 89 = 8 __ 9, the middle number is (8 + 9) = 17 so the answer is 979

11 x 47 = 4 __ 7, the middle number is (4 + 7) = 11 so the answer is 517

Harder trick:

11 x 223 = ?  2 _ _3, Write the two digits on the far left and right. Now add two numbers together,
starting with the unit digit. When the sum is  larger than 9,carry over to the next digit to the left.
2 _5(3+2) 3;  2 4 (2 + 2) 53 so the answer is 2453.

11 x 40532 = 4_ _ _ _ 2; 4_ _ _ 5 (2 +3) 2;  4_ _ 8(5+3)52;  4_ 5(0+5)852;  44(4 +0)5852,
so the answer is 445852  (This is fun!!)

Other interesting pattern:

11 x 11 =  121
111 x 111 = 12321
1111 x 1111 = 1234321
etc… till  111111111 x 111111111 = 12345678987654321

Divisibility rules for 11: 

The difference  of the sum of alternative digits is a multiple of 11, including “0”                                               (0 x 11 = 0, a multiple of 11.)

Example:

61985   (6 + 9 + 5) – (1 + 8) = 11   The number is divisible by 11.

7469     (7 + 6) – (4 + 9) = 0   The number is divisible by 11.

Try these mentally:(Answers below.)

#1: 11 x 23 =

#2: 11 x 72 =

#3: 11 x 97 =

#4: 11 x 55 =

#5: 11 x 76 =

#6: 11 x 60 =

#7: Sum of the first multiples of 11 smaller than 150.

#8:11 x 3421 =

#9: 11 x 452360 =

#10:  11 x 204673=

#11: 11111 x 11111=

#12: 1111111 x 1111111=

#13: What is the sum of the digits of 11111111 x 11111111?

#14: What is n if 45732n is divisible by 11?

#15: How many solutions for distinct numbers A and B if 4A8B is divisible by 11?

#16: How many solutions for distinct numbers A and B if A7B2 is divisible by 11?  What is their sum?

#17: How many solutions for distinct numbers A and B if A3B41 is divisible by 11?  

Answer key:
#1: 253
#2: 792
#3: 1067
#4: 605
#5: 836
#6: 660
#7 1001 The easiest way to do this is to see that this is an arithmetic sequence, starting with 11, 22, 33...
143 (11 x 13). There are 13 terms and the median is 77 so 13 x 77 or 13 x 7 x 11 = 1001
To do Mathcounts well, you need to know 7 x 11 x 13 =1001 by heart.
#8: 37631
#9: 4975960
#10: 2251403
#11:  123454321
#12:  1234567654321
#13:  The number is 123456787654321 so the sum of the digit is (4 x 7) x 2 + 8 = 64
#14:  n = 5
#15:  A + B = 12 (9, 3); (8, 4); (7, 5); skip (6, 6) because A and B are distinct (5, 7); (4, 8), (3, 9)
#16:  90, 81, 72, 63, 54, 45, 36, 27, 18;  A can’t be “0” so there are 9 pairs and the numbers are equally spaced, an arithmetic sequence. Thus the sum is median times how many numbers.                                      54 x 9 = 486
#17: A + B + 1 = 7 so A + B = 6 ; There are (6, 0);(5, 1); (4, 2); (2, 4); (1, 5) 
A + B + 1 = 7 + 11 = 18; A + B = 17; There are (9, 8) and (8, 9) so total 7 pairs

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

Polygon Part II: Interior/Exterior angles, Central Angles and Diagonals and Practice Problems

Learn the Basics here: Polygon Part I

Some important notes:

#1: Sum of all the interior angle of an n-sided polygon is \((n-2) * 180\)

#2: To find one interior angle of an n-sided regular polygon, you use :

\(\frac{(n-2)*180}{n}\)

or \({180 -\frac{360}{n}}\)", the latter will always give you one exterior angle of a regular n-gon.

#3:  Interior and its exterior angles are supplementary to each other. 

Interior angle A + exterior angle A = 180 degrees. 

#4: In every convex polygon, the exterior angles always add up to 360 degree.

Proof without words

#5: The central angle of a regular n-sided polygon is : \({\frac{360}{n}}\), same method as finding the exterior angle of a regular n-gon. Get more details on central angle here.

#6: Since \({\frac{360}{n}}\) will give you one exterior angle of a regular n-gon , 360 divided by one exterior angle of a regular n-gon will give you how many sides of that polygon. 

#7: To find how many diagonals an n-sided polygon has, you use: 

\(nC2 - n\) (Any two vertices except the sides will render one diagonal; however, order doesn't matter, thus choose 2.)

b. \( \frac{n(n-3)}{2}\) Any vertex, except its neighboring vertices and itself, can connect with other vertex to form a diagonal and there are n vertices; however, since order doesn't matter, AC is the same as CA so you divide the number by 2.

Questions to ponder: (Answer and solutions below.)

#1: The sum of the diagonals of two regular polygons is 44 and the sum of each of their interior angles is 264, what is the sum of their sides?

#2: If an exterior angle of a regular n-gon is 72, what is the measure of its interior angle? How many diagonals does that n-gon have? 

#3: What is one interior, exterior angle as well as how many diagonals are there for a 20 sided regular polygon?

#4: If each of the exterior angle of a regular polygon is 30, how many sides does that polygon have? 

#5: If the sum of all the interior angles of a polygon is 1440, how many sides does the polygon have? 

#6: How many degrees are there in the sum of a pentagon + a heptagon + a nonagon?

#7: The sum of the interior angles of a regular polygon is 720, what is the measure of one interior angle of that polygon? 

#8: 2000 Mathcounts State Target #8: (Check out Mathcounts here) : The total number of degrees in the sum of the interior angles of two regular polygons is 1980. The sum of the number of diagonals in the two polygons is 34. What is the positive difference between the numbers of sides of the two polygons?

                                                                               

# 9: Both pentagon and hexagon are regular. What is angle ABF?

#10:   B is the center of this regular pentagon. 

What is angle A, B and C? 

Answer key and solutions:

#1:  16: Learn the common polygon property by heart and check what the question is asked for.

In this case, let's see a few common polygons:

pentagon   5 sides     diagonals    5     interior angle  108 degrees

hexagon    6 sides     diagonals    9     interior angle  120  degrees

octagon     8 sides     diagonals   29    interior angle  135  degrees

nonagon    9 sides     diagonals   27    interior angle  140 degrees

decagon    10 sides   diagonals   35    interior angle  144 degrees

so the two polygons asked are hexagon and decagon, the sum of their sides are 6 + 10 = 16

#2: Interior and exterior angles are supplementary so 180 - 72 = 108 degrees  for its interior angle.

It's a pentagon and there are 5 diagonals in a pentagon.

#3: \(\frac{360}{20}\) = 18 degrees for the exterior angle

180 - 18 = 172 degrees for the interior angle of a 20-sided polygon.

#4: \(\frac{360}{30}\) = 12 sides 

#5:\(\frac{1440}{180}\) = 8 ; 8 + 2 = 10. It's a decagon (10 sides) -- when you find the sum of the interior angles you use (n - 2) * 180 so now you do the reverse.

#6: [ (5-2) + (7-2) + (9-2)] x 180 = (3 + 5 + 7) x 180 = 5 x 3 x 180 = 2700 degrees.

#7:\(\frac{720}{180}\)= 4 and 4 + 2 = 6 so this is a regular hexagon and one of its interior is \("\frac{720}{6}\) = 120 degrees. 

Or you can also do \(\frac{720 + 360}{180}\)

= 6 because all the interior angles and their exterior angles are supplementary and the sum of any exterior angle of a convex polygon is 360 degrees.

Using this method, you get how many sides instantly.

#8: This one is tricky because there are actually two different polygon pairs that the total number of degrees add up to 1980 degrees. One is a regular hexagon and a nonagon.  (720 + 1260)

Another is a heptagon and an octagon (900 + 1080). Looking at the sum of their diagonals, the latter is the right pair.  (14 + 20 = 34) so the answer is 8 - 7 = 1

#9: 360 - 108 - 120 = 132 degrees. 

#10: To find the central angle B, you do  \(\frac{360}{5}\)

= 72 degrees. BA and BC are both radius so the angle is

congruent. \(\frac{180-72}{2}\) = 54 degrees

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.