Problem Solving

Ian Stewart wrote:

bronzie35 wrote: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?

a. 5
b. 7
c. 11
d. 13
e. 17

Answer is c

Thank you for your help!

An even multiple of 15 is just a multiple of 30, so we want to find the largest prime factor of:

300 + 330 + 360 + 390 + ... + 570 + 600
= 30(10 + 11 + 12 + ... + 19 + 20)

There are a lot of ways to compute the sum of consecutive integers; for example, we can use the fact that the average of any 'equally spaced' list is always equal to the average of the smallest and largest numbers in that list. So the average of {10, 11, 12, ..., 19, 20} is just (10 + 20)/2 = 15. Since there are 11 numbers in the list, using:

avg = sum/n
sum = n*avg

the sum is equal to 11*15. So

300 + 330 + 360 + 390 + ... + 570 + 600
= 30(10 + 11 + 12 + ... + 19 + 20)
= 30*11*15
= 2*3*5*11*3*5
= 2*(3^2)*(5^2)*11

and the largest prime factor is 11.