GMAT Math

If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest prime factor of k?


The above question is from MGMAT CAT. The solution seems confusing.

The way I thought was as follows:

Multiples of 15 between 295 and 615 are all the multiples of 15 between 300 and 600


Sum = Avg * N

Avg: (Last + First)/2 = 600+300/2 = 450

N: (L-F)/Multiple + 1

(600-300)/15 + 1

300/15 + 1

= 21

Sum = 450 * 21

Prime Factors of 450 = 3.3.5.5.2 and Prime Factors of 21 = 7.3

Greatest Prime Factor = 7

Where did I go wrong?

It looks like your mistake was that you calculated the sum of all the multiples of 15 in that range, but the question just wanted the sum of all the even multiples of 15 (so all the multiples of 30). If you went for:

N = (600-300)/30 + 1

You should have had it. That tells you that there are 11 terms that average out to 450 for a sum of 4950, and then you can factor from there.

Really you had it...just one word off.

Seems a tad superfluous to multiply 450 by 11 only to have to factor it back again, given that 11 is the answer we're looking for.