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Question

Euclid's division lemma states that for two positive integers a and b, there exist unique integers q and r such that a=bq+r, where r must satisfy

A
1 < r < b
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B
0<rb
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C
0r<b
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D
0 < r < b
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Solution

The correct option is D 0r<b
If r must satisfy0r<b
Proof,
..,a3b,a2b,ab,a,a+b,a+2b,a+3b,..
clearly it is an arithmetic progression with common difference b and it extends infinitely in both directions.
Let r be the smallest non-negative term of this arithmetic progression.Then,there exists a non-negative integer q such that,
abq=r
=>a=bq+r
As,r is the smallest non-negative integer satisfying the result.Therefore, 0rb
Thus, we have
a=bq1+r1, 0r1b

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