Bunuel wrote:

Integer y equals the sum of all multiples of 21 between 210 and 441, inclusive. What is the greatest prime factor of y?

A. 7

B. 11

C. 19

D. 31

E. 37

We can first determine the sum of all multiples of 21 between 210 and 441 inclusive using the following formula:

sum = average x quantity

The quantity is (441 - 210)/21 + 1 = 231/21 + 1 = 11 + 1 = 12.

When we have a set of evenly spaced integers, we can calculate the average by using the following formula:

Average = (smallest multiple in the set + largest multiple in the set)/2. Thus:

Sum = (210 + 441)/2 x 12

Sum = 651/2 x 12 = 651 x 6

We need to determine the largest prime factor of 651 x 6.

Since we know the prime factors of 6, let’s prime factorize 651.

651 = 3 x 217 = 3 x 7 x 31. Thus, 31 is the greatest prime factor of y.

Alternative solution:

We are given that y = 210 + 231 + 252 + … + 441. We can factor 21 from each number. That is:

y = 21(10 + 11 + 12 + … + 21)

We can use the formula sum = average x quantity to calculate 10 + 11 + 12 + … + 21 as follows:

10 + 11 + 12 + … + 21 = [(10 + 21)/2] x 12 = 31/2 x 12 = 31 x 6

Thus y = 21(31 x 6) = 3 x 7 x 31 x 2 x 3.

So the largest prime factor of y is 31.

Answer: D

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Jeffery Miller

Head of GMAT Instruction

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