Related Articles

# GATE | GATE CS Mock 2018 | Question 1

• Last Updated : 09 Jan, 2018

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)

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.

Quiz of this Question

Attention reader! Don’t stop learning now.  Practice GATE exam well before the actual exam with the subject-wise and overall quizzes available in GATE Test Series Course.

Learn all GATE CS concepts with Free Live Classes on our youtube channel.

My Personal Notes arrow_drop_up