Problem Solving Strategy: Probability, Counting, Grid

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#5 1993 Mathcounts National Target : Find the probability that four randomly selected points on the geoboard below will be the vertices of a square? Express your answer as a common fraction.

#5 2004 AMC 10A: A set of three points is chosen randomly from the grid shown. Each three-point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?

Solution:

#5 National Target: There are 16C4 = (16 x 15 x 14 x 13)/ 4 x 3 x 2 = 1820 ways to select 4 points on the geoboard.

There are 3 x 3 = 9  one by one squares and 2 x 2 = 4 two by two squares and 1 x 1 = 1 three by three squares. (Do you see the pattern?)

                                                         

There are 4 other squares that have side length of √ 2
and 2 other larger squares that have side length of √ 5.

9 + 4 + 1 + 4 + 2 = 20 and 20/1820 = 1/91

#5 AMC-10A: There are 9C3 = (9 x 8 x 7) / 3 x 2 x 1 = 84 ways to chose the three dots and 8 of the lines connecting the three dots will form straight lines. (Three verticals, three horizontals and two diagonals.) so 

8/84 = 2/21

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

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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.