CAT Quantitative Aptitude Questions | CAT Number Systems - Remainders

The question is from CAT Number Theory - Remainders. A large number is given and we need to find out the remainder when the number is divided by 102. A range of CAT questions can be asked based on this simple concept. When we divide 24 pigeons into 5 groups, 4 are left out or remain. This is the remainder. When there is a question based on this idea in the exam, we have a CAT among the pigeons.

Question 14: If N = (24³ + 25³ + 26³ + 27³), then N divided by 102 leaves a remainder of?

1. 18
2. 12
3. 1
4. 0

Explanatory Answer

Method of solving this CAT Question from Number Theory - Remainders: What divides a³ + b³?

= (24³ + 25³ + 26³ + 27³)
Unit digits of N = 4 + 5 + 6 + 3 = 18
N is divisible by 2

We know that, a³ + b³ = (a + b)(a² – ab + b²)
=> (a + b) always divides a³ + b³
Therefore, 24³ + 27³ is divisible by 51 (24 + 27)

Also, 25³ + 26³ is divisible by 51 (25 + 26)
The given divisor 102 can be expressed as 51 * 2
Hence, N is completely divisible by 102
Remainder = 0

The question is "If N = (24³ + 25³ + 26³ + 27³), then N divided by 102 leaves a remainder of?"

Hence the answer is 0.

Choice D is the correct answer.