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Euclid’s division algorithm application to prove generalisedf results| Word problems

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This video demonstrates how to find HCF using Euclid’s division lemma or Euclid’s division algorithm.Statement of Euclid’s division algorithm and its appliction to solve word problems is illustratedin detail.Generalised results are proven by Euclid’s division algorithm.

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 .

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.

00:02 Q3. An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
01:10 use given condition
03:22 Analyse the question.Find HCF
03:36 Apply Euclid’s Division Algorithm to find HCF
03:42 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 is less than d .
03:55 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 .
04:42 Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF.

05:30 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.]
06:52 Euclid’s Division Algorithm
07:43 consider the case for r=0
08:41 Take r=1
10:14 consider the case for r=3
12:02 Result proved

12:12 Q5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9 m, 9 m + 1 or 9m + 8.
12:40 Note down the values of a , b in Euclid’s division lemma.
13:22 use Euclid’s division lemma
13:57 consider the case for r=0
14:48 Take r=1
16:43 consider the case for r=3

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