\(15600=2*2*3*13*2*5*2*5\)
A immediate though should no 3 consecutive integers will be multiple of 5, so both 5s have to be together
so one number is 25 or a multiple of 25
let us work on remaining factors 2*2*2*2*3*13... 13*2 will give us 26 and 2*2*2*3 will give 24
so 3 integers = 24*25*26
sum of square = \(24^2+25^2+26^2=(25-1)^2+25^2+(25+1)^2=25^2-2*25+1+25^2+25^2+2*25+1=3*25^2+2=3*625+2=1875+2=1877\)
D
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Hi,
If the product of three consecutive positive integers is 15600 then the sum of the squares of these integers is

A) 1777
B) 1785
C) 1875
D) 1877
E) 1897

Since, the product of 3 numbers is 15600, one of the numbers should be a multiple of 5. Since, 20^3 is 8000 and 30^3 is 27000, so one of the numbers should by 25. Join the dots and try to use POE (process of elimination) rather than trying to use algebra.

Now, make factors of 15600. You will get 2^4 x 3 x 5^2 x 13. Thus, one of the numbers should be a multiple of 13. So, it has to be 26.

Now, the 3 consecutive numbers could be either 24, 25, 26 or 25, 26, 27. We will pick 24, 25, 26 as 27 can not be part of the list. We have only one 3 in the factorization of 15600.

So, rather than using algebra, we used hit and trial to get the three consecutive numbers.


Now, you can either calculate the square the numbers and get the sum as 1877, which is going to be quite time consuming. Alternate way is to find the unit digit and then use ballparking.

Square of unit digit of 24 = 6
Square of unit digit of 25 = 5
Square of unit digit of 26 = 6

6 + 5 + 6 = 17
So, the unit digit of the resultant number should be 7. Eliminate B and C.
Now, find the square of 25 and multiply it with 3. You will get 1875. So, the answer should be close to 1875.

Thus, the answer is D.

Please give kudos if you liked the solution.


Deepti Singh
Master Trainer - GMAT
Manya - The Princeton Review
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In this, the options are sufficient to get almost to the answer. Only a small amount of calculations are required in the end.


We know that 15600 is a perfect cube who's cube root lies between 20 (\(20^3 = 8000\) and 30 (\(30^3 = 27000\)).


We also know that since 15600 ends in 2 zeroes, there has to be 2 fives in these 3 numbers (as a 0 is a product of a 2 and a 5), therefore one of the numbers is 25.


So the 3 numbers could be (a) 23, 24, 25 or (b) 24, 25, 26 or (c) 25, 26, 27

Now a glance at the options tells us that the sum of squares is an odd number. (All the options are odd)



If 2 numbers are Odd and the 3rd is even, then the resulting number will be even (\(O^2 + O^2 + E^2 = O + O + E = E + E = E)\), so (a) and (c) are ruled out and the numbers are therefore 24, 25 and 26.


Sum of squares \(24^2 + 25^2 + 26^2 = 576 + 625 + 676 = 1877\)



Option D

Arun Kumar
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I did the factors of 15600=2*2*2*2*3*5*5*13

Next I analysed that 13 is biggest prime and should be multiply by smallest prime keeping in mind that we have all 3 numbers as consecutive integers.

Also we know that there should numbers that have last digit and 5 and multiple of 2 to get a hundred at the end.

So from factors-->2*2*2*2*3*5*5*13
I took out 13*2=26
5*5=25
2*2*2*3=24

So we got 2 consecutive integers-->24,25,26

Next is sum of squares of this number=576+625+676=1877

Therefore D.
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The product of three consecutive integers is 15600.

One of the numbers will end with '5' as unit digit as we need '0' in unit digit because then number 15600 lies between [\((20)^3\)= 8000 and \((30)^3\) = 27000].

=> With a hit n trial we will get those three integers as 24, 25, and 26.

=> \((24)^2 + (25)^2 + (26)^2\) = 576 + 625 + 676

=> 1877

Answer D
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We need to sum the squares of three consecutive integers. If those integers are a-1, a and a+1, we need to find

(a-1)^2 + a^2 + (a + 1)^2 = a^2 - 2a + 1 + a^2 + a^2 + 2a + 1 = 3a^2 + 2

So the right answer must be 2 greater than some multiple of 3 (i.e. it must give a remainder of 2 when we divide it by 3). If you sum the digits of any positive integer at all, and divide that sum by 3, you'll always find the remainder you'd get when you divide the original number by 3. Here, if we add the digits of each answer choice, only answer D gives us a remainder of 2 when we divide by 3, so it is the only answer that could be correct
.
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\((n -1)n(n+1) = 15600\)

\(24*25*26 = 15600\)

So, \(n = 25\)

Sum of the squares of the 3 numbers is -

\(24^2 + 25^2 + 26^2 = 1877\). Answer must be (D)
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