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 = 55^{5} + 17^{5} – 72^{5}, then which of the following is true?

- N is divisible by both 7 and 13
- N is divisible by both 3 and 17
- N is divisible by 17 but not 3
- N is divisible by 11 but not 17

N is divisible by both 3 and 17

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N = 55^{5} + 17^{5} – 72^{5}

N = (54 + 1)^{5} + (18 - 1)^{5} – 72^{5}

The above is divisible by 3

N = (51 + 4)^{5} + 17^{5} – (68 + 4)^{5}

The above is divisible by 17

N = 55^{5} + (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?"**

Choice B is the correct answer.

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