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tricotasu

Answered question

2021-01-31

Prove the transitivity of modular congruence. That is, prove that for all integers a,b,c, and n with n > 1, if

a = b (m o d n) a n d b = c (m o d n) t h e n a = c (m o d n)

firmablogF

Skilled2021-02-01Added 92 answers

The transitivity of modular congruence is that for all integers a, b, c and n with n > 1, if

a \equiv b (m o d n) a n d b \equiv c (m o d n) t h e n a \equiv c (m o d n)

a \equiv b (m o d n) a n d b \equiv c (m o d n)

, then there exists integers k and k’ such that,

q - b = n k

b - c = n k^{'}

Adding these two equations yields

a - c = n (k + k^{'})

where, k +k’ is an integer.
And so

a \equiv c (m o d n)

Conclusion:
The theorem of transitivity of modular congruence is proved.

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