GATE CS Mock Tests-1 | Buy Now |

1.

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

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

- (7^700) mod 100 = 7^(700 mod 100) = 7^0 = 1.
- 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.
- (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