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Let k be the largest integer such that 343/7^k is an integer. What is the remainder when 343/7^k is divided by 7

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4 votes

Answer:

1

Explanation:

The given ratio will be the remaining factor of 343 after factors of 7 have been removed. The value of interest is that factor, modulo 7.

Factors of 343

343 = 7·7·7 = 7³

Ratio

The value of k is the power of 7 in the factorization of 343, so we have ...

k = 3

and ...

343/7³ = 1

Remainder

The remainder when this ratio is divided by 7 is ...

1 mod 7 = 1

The remainder when 343/7^k is divided by 7 is 1.

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