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 divisibility of 7, 11, 13, 17. A range of CAT questions can be asked based on this simple concept of Remainders. 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 12: Given a number, N = 55⁵ + 17⁵ – 72⁵, then which of the following is true?

1. N is divisible by both 7 and 13
2. N is divisible by both 3 and 17
3. N is divisible by 17 but not 3
4. N is divisible by 11 but not 17

Explanatory Answer

Method of solving this CAT Question from Number Theory - Remainders: Try expressing the given number in terms of multiples of the divisors.

N = 55⁵ + 17⁵ – 72⁵
N = (54 + 1)⁵ + (18 - 1)⁵ – 72⁵
The above is divisible by 3

N = (51 + 4)⁵ + 17⁵ – (68 + 4)⁵
The above is divisible by 17
N = 55⁵ + (11 + 6)⁵ – (66 + 6)⁵
The above is divisible by 11
N cannot be expressed in favourable of 7 and 13 and hence is not divisible by 7 and 13
From the above the best answer choice is (B) – N is divisible by both 3 and 17

The question is "which of the following is true?"

Hence the answer is N is divisible by both 3 and 17.

Choice B is the correct answer.