Question

Using Euclid's algorithm, find the HCF of 1651 and 2032. Express
the HCF in the form (1651m + 2032n) for some integers m and n.​

Answer 1

Step-by-step explanation:

We know, Euclid algorithm lemma , a = bq + r where 0 ≤ r < b

From Euclid algorithm lemma , 2032 = 1651 × 1 + 381

again, using lemma for 1651 and 381 , e.g., 1651 = 381 × 4 + 127

similarly use lemma for 381 and 127 , e.g., 381 = 127 × 3 + 0

Hence, HCF = 127

Now, 127 = 1651M + 2032N

⇒127 = (127 × 13)M + (127 × 16)M

⇒1 = 13M + 16N

Here many solutions possible because we have one equation contains two variable .

If I assume M = 5 and N = -4

Then, 13 × 5 + 16 × -4 = 65 - 64 = 1

So, HCF of 1651 and 2032 in the form of 1651M + 2032N is [1651(5) + 2032(-4)]

3.9

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