What is the remainder when 5^{99} is divided by 13? — GMAT

What is the remainder when $5^{99}$ is divided by $13$ ?

Answer: 8

The remainder when

5^{99}

is divided by

13

8

To find the remainder of

5^{99}

divided by

13

, find the repeating pattern of remainders for powers of

5

modulo

13

. The first four powers yield remainders

5, 12, 8,

and

1

. Since

5^4 \equiv 1 \pmod{13}

, the remainders repeat every

4

powers. Dividing the exponent

99

4

gives

99 = 4 \times 24 + 3

. The remainder of

3

indicates that

5^{99}

has the exact same remainder modulo

13

5^3

, which is

8

Analyze the remainders of powers of 5 modulo 13 to identify the repeating period length.

5^1 \equiv 5 \pmod{13}

5^2 \equiv 12 \pmod{13}

5^3 \equiv 8 \pmod{13}

, and

5^4 \equiv 1 \pmod{13}

. The sequence of remainders repeats every 4 powers.

Integral powers modulo a positive integer exhibit periodic behavior.

Compute the remainder of the exponent 99 divided by the cycle length 4.

99 \div 4 = 24

with a remainder of

3

The exponent's remainder modulo the cycle length determines which element in the periodic cycle gives the equivalent value.

Evaluate the 3rd term in the remainder cycle.

The 3rd element in the sequence

(5, 12, 8, 1)

8

An exponent congruent to

3 \pmod 4

yields the same remainder as

5^3 \pmod{13}

Modular Arithmetic and Cyclicity of Powers

Question