What is the smallest positive integer m such that the remainder when 6^m is divided by 11 … — GMAT

What is the smallest positive integer $m$ such that the remainder when $6^m$ is divided by $11$ is equal to $8$ ?

Answer: 7

The smallest positive integer

m

is 7.

Evaluating powers of 6 modulo 11 sequentially gives:

6^1 \equiv 6

6^2 \equiv 3

6^3 \equiv 7

6^4 \equiv 9

6^5 \equiv 10

6^6 \equiv 5

, and

6^7 \equiv 8 \pmod{11}

. Therefore,

m = 7

is the smallest positive integer satisfying the condition.

Calculate the remainder of powers of 6 when divided by 11 starting from

m = 1

6^1 \equiv 6 \pmod{11}

6^2 \equiv 3 \pmod{11}

6^3 \equiv 7 \pmod{11}

6^4 \equiv 9 \pmod{11}

6^5 \equiv 10 \pmod{11}

6^6 \equiv 5 \pmod{11}

6^7 \equiv 8 \pmod{11}

Modular multiplication allows computing powers step-by-step by multiplying the previous remainder by 6 and finding the remainder modulo 11.

Identify the smallest exponent

m

that yields a remainder of 8.

m = 7

m = 7

6^7 = 279,936 = 11 \times 25,448 + 8

, which leaves a remainder of 8.

Modular Arithmetic and Cyclicity of Powers

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