Prime Factors

Translation Task 1

We can translate statements about numbers into ones about prime factors. For example, “35 is a factor of 210” means the same as “all the prime factors of 35 are also prime factors of 210”. Alternatively, “210 contains the prime factors of 35”. Translate the following sentences into ones about prime factors. (answers below)

1. 315 is the lowest common multiple of 45 and 21.
2. 15 is the highest common factor of 660 and 315
3. x is a multiple of y
4. P is a perfect square

Official Guide practice (find these in your Official Guide or at www.gmatclub.com)

PS01248 Bouquets are to be made using white tulips and red tulips

If the positive integer x is a multiple of 4

PS02704 If n = 20! + 17,

PS10982 If n is a positive integer and the product

PS00041 If p is the product of the integers from 1 to 30, inclusive

PS12177 Club X has more than 10 but fewer than 40 members

PS05882 If y is the smallest positive integer such that 3,150

PS06498 How many prime numbers between 1 and 100

PS08209 For the positive integers a, b, and k,

Translation Task 2

Here are some more advanced sentences to translate into ones about prime factors. (answers below)

1. e/d is an integer, and f/e is an integer
2. R is a perfect cube
3. Is is a divisor of x?
4. Z is a factor of (X+Y)
5. Are m and n both factors of s?
6. a, b, and c are integers and ab = 3c

Translation Task 1 answers

1. 315 is the smallest number that contains the prime factors of 45 (5, 3, 3) and 21 (3, 7). The prime factors of 315 are 3, 3, 5, 7
2. 15 contains the common prime factors of 660 (2, 2, 3, 5, 11) and 315 (3, 3, 5, 7). The prime factors of 15 are 3, 5.
3. “x is a multiple of y” means means x contains all the prime factors of y
4. “P is a perfect square” means the prime factors of P can be divided into two identical groups

Translation task 2 answers

1. ‘e/d is an integer, and f/e is an integer’ means f contains all the prime factors of d. In the prime factor tree of f, there will be a branch containing e, and from there a branch containing d. Note also that the statement ‘e/d is an integer’ means d is a factor of e.
2. ‘R is a perfect cube’ means the prime factors of R can be divided into three equal groups.
3. ‘Is is a divisor of x?’ means Does x contains k number of 2s as prime factors? Note that ‘divisor’ means the same as ‘factor’
4. ‘Z is a factor of (X+Y)’ means (X+Y) contains all the prime factors of Z. If you thought that this means that Z is a factor of X and Y individually, you’re mistaken. Think up an example to check this.
5. ‘Are m and n both factors of s?’ means does s contains the prime factors of m and the prime factors of n. However, this is not the same as asking if s contains all the prime factors of mn, as there may be some “overlap”. Consider the example m=10, n=15 and s=90
6. ‘a, b, and c are integers and ab = 3c’ means a or b must contain at least one 3 as a prime factor and c must be made up of the remaining prime factors of a and b. The prime factors on both sides of this equation must balance out.