NDA Mathematics Question (2025) — Solution

Step-by-step solution

To find the remainder when 5^ 99 is divided by 13, we evaluate the powers of 5 modulo 13. 5^1 5 13 5^2 = 25 -1 13 Squaring both sides gives: 5^4 (-1)^2 1 13 We can express 5^ 99 in terms of 5^4 as follows: 5^ 99 = 5^ 96 5^3 = (5^4)^ 24 125 Substituting 5^4 1 13 : 5^ 99 (1)^ 24 125 13 5^ 99 125 13 Dividing 125 by 13 yields 125 = 13 9 + 8. Thus, 125 8 13 . The remainder is 8. Answer: 8