NCERT Solutions for Maths CBSE Class 10 Real Numbers Ex 1.1 Q4
Euclid’s Division Lemma : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 = r is less than b.
A lemma is a proven statement used for proving another statement.
Euclid’s division algorithm is a technique to compute the Highest Common Factor HCF of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b .
00:02 Q4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3 m + 1 for some integer m. [ Hint : Let x be any positive integer then it is of the form 3 q, 3 q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3 m + 1.]
01:22 Euclid’s Division Algorithm
02:13 consider the case for r=0
03:11 Take r=1
04:34 consider the case for r=3
06:32 Result proved
Euclid’s division algorithm
To find the HCF of two positive integers, say c and d, with c is greater than d, followthe 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 is less than d .
Step 2 : If r = 0, d is the HCF of c and d . If r not zero 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.
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