Wednesday, May 9, 2012

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.

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