Sum of five consecutive integers: n+(n+1)+(n+2)+(n+3)+(n+4)=5n+10=5(n+2)

Sum of middle terms: (n+1)+(n+2)+(n+3)=3n+6=3(n+2)

If 5(n+2) is a perfect cube, then the value of n+2 could be 5^2, 5^5, 5^8,...
If 3(n+2) is a perfect square, then the value of n+2 could be 3, 3^3, 3^5,...

In orded to satisfy both conditions, the minimum value of n+2 should be (5^2)(3^3)=(15^2)*3=225*3=675

c=n+2=675
Ans. D

Posted from my mobile device

Our numbers are consecutive integers, so are c-2, c-1, c, c+1 and c+2. So we know 3c is a perfect square and 5c is a perfect cube. But in a perfect square, the exponents in your prime factorization are all even, so c will need to be divisible by at least one more 3. And in a perfect cube, your exponents in your prime factorization are all multiples of 3, so if 5c is a perfect cube, c itself will need to be divisible by at least 5^2.

Since we want the smallest value of c, we don't want c to be divisible by any unnecessary primes, so c = 3^a * 5^b. Since 3c is a perfect square, a+1 and b must be even, so a is odd and b is even. Since 5c is a perfect cube, the exponents in 3^a * 5^(b+1) must be multiples of 3, so a is divisible by 3, and b is one less than a multiple of 3. From all of that, the smallest positive value of a is 3, and the smallest positive value of b is 2, and c = (3^3)(5^2) = 675.
_________________

I solved with this method, except that I chose 3 to be the minimum number and multiplied 3*5^2. It is easy to see that 5^2 is the minimum number required, but how is 3^3 the minimum number that satisfies the 2nd condition ?

Will appreciate your response.

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________