According to Euclid’s division algorithm, using Euclid’s division lemma for any two positive integers a and b with a > b enables us to find the
$
  (a) {\text{ H}}{\text{.C}}{\text{.F}} \\
  (b) {\text{ L}}{\text{.C}}{\text{.M}} \\
  (c) {\text{ Decimal expansion}} \\
  (d) {\text{ Probability}} \\
$

Hint: In this question use the basic definition of Euclid’s division lemma and apply it for a special condition that is when the unique integer to be found and which satisfies the division lemma is taken as zero.

Complete step-by-step answer:

According to Euclid’s division lemma,
For each pair of positive integers a and b, we can find unique integers p and q satisfying the relation
$a = bp + q$, where $0 \leqslant q \leqslant b$
So if q = 0 then p is the H.C.F of a and b.
The basis of the Euclidean division algorithm is Euclid’s division lemma.
To calculate the highest common factor (H.C.F) of two positive integers a and b we use Euclid’s division algorithm.
H.C.F is the largest number which exactly divides two or more positive integers.
By exactly we mean that on dividing both the integers a and b the remainder is zero.
So this is the required answer.
Hence option (A) is correct.

Note: In questions there may be a term called Euclid’s division algorithm, we must not confuse between Euclid’s division lemma and Euclid’s division algorithm as both are different. Lemma is a proven statement which is used for providing another statement whereas algorithm is a series of well-defined steps which gives a procedure of solution.