Lemma :Lemma is a proven statement which is used to prove another statement.

Algorithm :Algorithm is a series of well defined steps which gives a procedure for solving a type of problem.

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Euclid's lemma : This lemma is nothing but a restatement of the long division process.
Let a and b be any two positive integers. Then, there exist unique integers q and r
Such that: a = bq + r, o < = r < b

a = dividend
b = divisor
q = quotient
r = remainder
Let the division of one positive integer by another say 60 by 7 . The division can be carried out as
follows : 60 = 7 x 8 + 4

o < = 4 < 7

Euclid's Division Algorithm :

Euclid's Division Algorithm is also an sequential process (Algorithm) to compute the highest common factor (HCF) or greatest common division of two given positive integers.

If a and b are positive integers such that a = bq+r, then every common divisor of a and b is a common divisor of b and r, and vice-versa.

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e.g. Use Euclid's division algorrithm to find the HCF of 220 and 65.
Solution : Given integers are 220 and 65, clearly 220 > 65 . Now apply Euclid lemma
We get,
220 = 65 x 3 + 25
again apply the same lemma
65 = 25 x 2 + 15
25 = 15 x 1 + 10
15 = 10 x 1 + 5
10 = 5 x 2 + 0
H.C.F is 5.