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Q.

Euclid's Division Lemma states that, if two positive integers ‘a’ and ‘b’, then there exists unique integers ‘q’ and ‘r’ such that which satisfies the condition:
[[1]] where $0 \leq r \leq b$ .

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answer is A = BQ + R.

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Detailed Solution

Euclid's Division Lemma states that, if two positive integers ‘a’ and ‘b’, then there exists unique integers ‘q’ and ‘r’ such that which satisfies the condition:

a = bq + r

where

0 \leq r \leq b

.
It is used to calculate the highest common factor between any two integers.
Thus, if there are two integers, say ‘a’ and ‘b’ then by normal division, it is possible to represent the larger one in terms of the smaller one as

a = bq + r

where ‘q’ is the quotient, ‘r’ is the remainder and

0 \leq r \leq b

.
This procedure continues by making the remainder, the new divisor and the original divisor as the new dividend, until

r = 0

.
Therefore, when

r = 0

then that respective divisor is the H.C.F of the two integers, a and b in this case.

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