Problem Solving

lenagmat wrote:Integer m has 4 different prime factors and n has 3 different prime factors. If m and n has the greatest common factor of 15, how many different prime factors does mn have?

As the GCF of m and n is 15 = 3*5, m and n both must have 3 and 5 as their prime factors. Hence, the other two prime factors of m and the third prime factor n must be different from each other.

Hence, the different prime factors of mn are 3, 5, two other prime factors of m and third prime factor of n.

Hence, number of different prime factors of mn = 2 + 2 + 1 = 5

The correct answer is B.

lenagmat wrote:Integer m has 4 different prime factors and n has 3 different prime factors. If m and n has the greatest common factor of 15, how many different prime factors does mn have?

A). 4
B). 5
C). 6
D). 7
E). 8

Alternatively, we could pick some numbers that satisfy the given conditions.
As was already pointed out, we know that m and n must share a 3 and a 5 in their prime factorizations.
So, one possibility is: m=2x3x5x7 and n=3x5x11

This means that mn = 2x2x3x5x5x7x11
Here we can see that mn has 5 different prime factors. They are 2, 3, 5, 7, and 11

So, the answer is B

Cheers,
Brent

Made a silly mistake.
Edited the reply.

lenagmat wrote:Integer m has 4 different prime factors and n has 3 different prime factors. If m and n has the greatest common factor of 15, how many different prime factors does mn have?

A). 4
B). 5
C). 6
D). 7
E). 8

If m and n have a GCF of 15, they both share a prime of 3 and 5. Thus, m has 2 prime factors that differ from those of n and n has 1 other prime factor that differs from those of m.

Since mn has 2 common prime factors and 3 uncommon prime factors, mn has a total of 5 different prime factors.

Answer: B