Prime Factors

Translate the following sentences into ones about prime factors. The first one is done for you. (Answers below)

1. x is a factor of y means all the prime factors of x are prime factors of y
2. A is the lowest common multiple of B and C
3. Is p is a multiple of q?
4. u is the highest common factor of v and w
5. e/d is an integer, and f/e is an integer
6. R is a perfect cube
7. Is 2^k a divisor of x?
8. Z is a factor of (X+Y)
9. Are m and n both factors of s?
10. a, b, and c are integers and ab = 3c

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1. x is a factor of y means all the prime factors of x are prime factors of y
2. ‘A is the lowest common multiple of B and C’ means A is composed of the minimum group of prime factors that contains all the prime factors of B and all the prime factors of C. It might help to create a Venn diagram of the prime factors of B and C to solve this.
3. ‘Is p is a multiple of q?’ means Does p contain all the prime factors of q? This is the same as asking ‘Is q a factor of p?’.
4. ‘u is the highest common factor of v and w’ means u contains all the prime factors that v and w share and no more. It might help to create a Venn diagram of the prime factors of v and w to solve this.
5. ‘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.
6. ‘R is a perfect cube’ means the prime factors of R can be separated into three equal groups.
7. ‘Is 2^k a divisor of x?’ means Does x contains k number of 2s as prime factors? Note that ‘divisor’ means the same as ‘factor’
8. ‘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.
9. ‘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
10. ‘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.