### Number Theory

**Question**

What is the remainder
when (13

^{100}+1 7^{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 = 2
Remainder of dividing
2

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

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

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

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

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

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

^{101}by 25 = Remainder of dividing 2^{100}by 25 * Remainder of dividing 2^{1}by 25 = 1 * 2 = 2Labels: 2IIM, 2iim Chennai, CAT 2013 questions, CAT 2013 Solutions, CAT number theory, CAT remainders, Fermat's Little Theorem

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