If N = 24^3 + 25^3 + 26^3 + 27^3, then N divided by 102 leaves a remai

Question

If  N = 24^3 + 25^3 + 26^3 + 27^3 , then N divided by 102 leaves a remainder of?

A. 18
B. 12
C. 2
D. 1
E. 0

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GMATinsight · Accepted Answer

N = 24^3 + 25^3 + 26^3 + 27^3

24+25+26+27 = 102

i.e. when  N = 24^3 + 25^3 + 26^3 + 27^3  divided by 102, remainder will be 0

Answer: Option E

ArunSharma12 · Answer

formula in use:  a^3+b^3 = (a+b)(a^2-ab+b^2)

102 = 2*51
 24^3+27^3 = (51)(24^2-24*27+27^2) = 51 * odd  
 24^2-24*27+27^2 = (27-24)^2 + 24*27 = 9 + even = odd

25^3+26^3 = (51)(25^2-25*26+26^2) = 51 * odd 
 25^2-25*26+26^2 = (26-25)^2 + 26*25 = 1 + even = odd

24^3+27^3 + 25^3+26^3= (51)(24^2-24*27+27^2) + (51)(25^2-25*26+26^2)= 51 (odd + odd)= 51 * even   
 \frac{24^3+27^3 + 25^3+26^3}{102}= \frac{51 * even }{102}= 0 
Ans: E

nick1816 · Answer

Sum of 2 odd numbers and 2 even numbers is even.
Hence N is divisible by 2

a^3+b^3 is always divisible by a+b

+  = 51a+51b

N divided by 102 leaves a remainder of 0

GaurSaini · Answer

N=24^3+25^3+26^3+27^3

a^n + b^n is divided by (a+b) if n is odd.

Here we can see that power of numbers is odd, so we will try to look onto them as per the divisor '102'.

102 = 2 * 51

24+27 = 51
25+26 = 51

Hence we will try to convert the given numbers in form of multiples of '51'. As we know, (a^3 + b^3) = (a+b)(a^2+b^2+a*b)

24^3 + 27^3 = (24+27)(24^2+27^2+24*27) = 51*odd
25^3 + 26^3 = (25+26)(25^2+26^2+25*26) = 51*odd

51*odd + 51*odd = 51(odd+odd) = 51*even = 102*number

Hence N=24^3+25^3+26^3+27^3 is divided by 102 and remainder will be '0'.

sujoykrdatta · Answer

We know that (a^3 + b^3) is divisible by (a + b)
24^3 + 27^3 is divisible by 24 + 27 i.e. 51
25^3 + 26^3 is divisible by 24 + 27 i.e. 51
Thus, 24^3 + 25^3 + 26^3 + 27^3 is divisible by 51
Also, the above sum is even (2 even and 2 odd numbers) and hence divisible by 2

Thus, 24^3 + 25^3 + 26^3 + 27^3 is divisible by 51 and 2 i.e., by 102
Thus, remainder is 0
Answer E

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If N = 24^3 + 25^3 + 26^3 + 27^3, then N divided by 102 leaves a remai

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If N = 24^3 + 25^3 + 26^3 + 27^3, then N divided by 102 leaves a remai

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