Good evening sir/ma'am.
For finding HCF by Euclid algorithm, is it necessary to do division for finding quotient and remainder?

Euclid's Division Algorithm (Lemma)
We know that how to divide a positive integer 'a' by another positive integer 'b' by long division method.

Let us consider a = 217 , b = 5 and make the division of 217 by 5 as under :

Dividend

?

Divisor ? 5)217(43 ? Quotient

20

--------

17

15

-----

2 ?Remainder

Clearly, 217 = 5 x 43 + 2

i, e.,

Dividend = Divisor ? Quotient + Remainder
Let us consider some more pairs of integers as given below :

(i). a = 191; b = 2; 191 = 2 x 95 + 1

(ii).a = 321; b = 3; 321 = 3 x 107+0

(iii).a = 205; b = 7; 205 = 7 x 29 + 2

(iv). a = 1047; b = 11; 1047=11 x 95+ 2

From the above, we conclude that for each pair of positive integers, a and b, there exists a pair of whole numbers q and r such that a = b ? q + r, 0 ? r
Formal statement of Euclid's Division Lemma
For any two given positive integers a and b, there exist unique integers q and r such that a = bq + r, 0 ? r

yes dhairya it is important to do division for finding quotient and remainder...
the above answer with example is absolutely correct...