**Question**

What is the remainder when (13

^{100}+17^{100}) is divided by 25?A. 2

B. 0

C. 15

D. 8

**Correct Answer: Choice (A)**

**Explanation:**What is the remainder when (13

^{100}+1 7^{100}) is divided by 25?(13

^{100}+1 7^{100}) = (15 – 2)^{100}+ (15 + 2)^{100}Now 5

^{2}= 25, So, any term that has 5^{2}or any higher power of 5 will be a multiple of 25. So, for the above question, for computing remainder, we need to think about only the terms with 15^{0}or 15^{1}.(15 – 2)

^{100}+ (15 + 2)^{100}Coefficient of 15

^{0}= (-2)^{100}+ 2^{100}Coefficient of 15

^{1}=^{100}C_{1}* 15^{1}* (-2)^{99}+^{100}C_{1}* 15^{1}* (-2)^{99 }. These two terms cancel each other.So, the sum is 0.Remainder is nothing but (-2)

^{100}+ 2^{100 }=(2)^{100}+ 2^{100}2

^{101}Remainder of dividing 2

^{1}by 25 = 2Remainder of dividing 2

^{2}by 25 = 4Remainder of dividing 2

^{3}by 25 = 8Remainder of dividing 2

^{4}by 25 = 16Remainder of dividing 2

^{5}by 25 = 32 = 7Remainder of dividing 2

^{10}by 25 = 7^{2}= 49 = -1Remainder of dividing 22

^{0}by 25 = (-1)^{2}= 1Remainder of dividing 2

^{101}by 25 = Remainder of dividing 2^{100}by 25 * Remainder of dividing 2^{1}by 25 = 1 * 2 = 2