Question

Difficulty: EasyModular Arithmetic

Determine the least non-negative integer congruent to $68 \pmod{9}$ .

Answer: 5

The least non-negative integer congruent to

68 \pmod{9}

is 5.

Dividing 68 by 9 gives a quotient of 7 and a remainder of 5. Since

0 \le 5 < 9

, the value 5 is the standard non-negative remainder.

Divide 68 by the modulus 9 using the division algorithm.

68 = 9 \times 7 + 5

Every integer

a

can be written uniquely as

a = nq + r

where

q

is the quotient and

0 \le r < n

Extract the non-negative remainder

r

r = 5

In modular arithmetic, the least non-negative integer congruent to

a \pmod{n}

is the remainder

r

when

a

is divided by

n

Modular Arithmetic Remainder

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