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Find a formula for the integer with smallest absolute value that is congruent to...

Apr 8, 2024

Find a formula for the integer with smallest absolute value that is congruent to an integer a modulo m, where m is a positive integer.

Solution by Steps

step 1

To find the smallest absolute value integer congruent to

a

modulo

m

, we first calculate

a \mod m

step 2

Let

a \mod m = r

, where 0 \leq r < m . This

r

is the remainder when

a

is divided by

m

step 3

The integers congruent to

a

modulo

m

are of the form

a + km

, where

k

is an integer

step 4

To minimize the absolute value, we choose

k

such that

|a + km|

is minimized

step 5

r \leq \frac{m}{2}

, then the smallest absolute value is

r

itself. If r > \frac{m}{2} , then the smallest absolute value is

r - m

step 6

Therefore, the formula for the integer with the smallest absolute value congruent to

a

modulo

m

r

r \leq \frac{m}{2}

, otherwise

r - m

Answer

The integer with the smallest absolute value congruent to

a

modulo

m

r

r \leq \frac{m}{2}

, otherwise

r - m

, where

r = a \mod m

Key Concept

Modular Arithmetic and Absolute Value

Explanation

In modular arithmetic, to find the integer with the smallest absolute value congruent to

a

modulo

m

, we calculate

a \mod m

and adjust based on whether the remainder is less than or greater than half of

m