Real Numbers - Class X |

Formulae, Relations, Theorems etc. |

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Lemma - A lemma is a proven statement used for proving another statement.

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

Euclid's Division Lemma - Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Eulid's Division Algorithm - So, let us state Euclid’s division algorithm clearly.To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q andr such that c = dq + r, 0 ≤ r < d.Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.Step 3 : Continue the process till the remainder is zero. The divisor at this stage willbe the required HCF.

Lemma - A lemma is a proven statement used for proving another statement.

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

Euclid's Division Lemma - Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Eulid's Division Algorithm - So, let us state Euclid’s division algorithm clearly.

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1 : Apply Euclid’s division lemma, to c and d. So, we find whole numbers, q and

r such that c = dq + r, 0 ≤ r < d.

Step 2 : If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.

Step 3 : Continue the process till the remainder is zero. The divisor at this stage will

be the required HCF.