GATE | GATE CS Mock 2018 | Question 1

Which of the following statement(s) is/are correct?

1. 1 is the remainder when 7^700 is divided by 100.
2. 1 is the remainder when 7^26 is divided by 100.
3. 2 is the remainder when 7^35 is divided by 13.

(A) Only (1)
(B) Only (2)
(C) Only (1) and (3)
(D) All (1), (2), and (3)


Answer: (C)

Explanation:

1. (7^700) mod 100 = 7^(700 mod 100) = 7^0 = 1.
2. Last two digits of 7^1 are 07
   Last two digits of 7^2 are 49
   Last two digits of 7^3 are 43
   Last two digits of 7^4 are 01.
   This cycle is 7, 9, 3, 1, 7, 9, …

   And, 26 mod 4 = 2.
   Therefore, 7^26 = 7^(4n+2) will end in 49.
   → 7^26 mod 100 = 7^2 mod 100 = 49.

3. (7^35) mod 13 = ((7^12)*(7^12)*(7^11)) mod 13

   A^(P-1)/P (where A is any natural number, and P is any prime number which is not a factor of A) will give remainder 1.
   Therefore,
   (7^12) mod 13 = 1
   So,
   → (1*1*(7^11)) mod 13
   → (7^11) mod 13
   → ((7)*(7^10)) mod 13
   → ((7)*((7^2)^5)) mod 13
   → ((7)*(49^5)) mod 13
   → ((7)*((39+10)^5)) mod 13
   → ((7)*(10^5)) mod 13
   → ((7)*(100*100*10)) mod 13
   → ((7)*(9*9*10)) mod 13
   → 2

Therefore, only statements (1) and (3) are correct.
Option (C) is true.

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