List the first 5 positive numbers that are congruent to 5 in modulo 13

Mafia We
2022-06-05
Answered

List the first 5 positive numbers that are congruent to 5 in modulo 13

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Haidar Ali

Answered 2022-07-21
Author has **4** answers

Let x be a positive number that is congruent to 5 in module 13. Mathematically it can be written as:

$x\equiv 5mod\left(13\right)$

Note that a number x is congruent to 5 mod 13 if and only if x-5 is divisible by 13. It follows that the possible positive numbers for which x-5 is divisible by 13 are as follows:

5,18,31,44,57, . . .

One can check that:

(1) 5-5=0 and 0 is divisible by 13. This shows that:

$5\equiv 5mod\left(13\right)$

(2) 18-5=13 and 13 is divisible by 13. This shows that:

$18\equiv 5mod\left(13\right)$

(3) 31-5=26 and 26 is divisible by 13. This shows that:

$31\equiv 5mod\left(13\right)$

(4) 44-5=39 and 39 is divisible by 13. This shows that:

$44\equiv 5mod\left(13\right)$

(5) 57-5=52 and 52 is divisible by 13. This shows that:

$57\equiv 5mod\left(13\right)$

The first five positive numbers that are congruent in 5 modulo 13 are:

5,18,31,44,57

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