Question

1. Use Euclid's Division Algorithm to find HCF the
following:

i) 867 and 255

ii) 135 and 225

iii) 616 and 32

iv) 455 and 42

2. What is Euclid's Division Lemma? Explain with the
help of suitable example.

3. What is Euclid's Division Algorithm?​

Answer 1

Answer:

 1.  i)According to the definition of Euclid's theorem,

a=b×q+r where 0≤r<b.

Now,  

867 and 255

867>255 so we will divide 867 by 225

867=255×3+102

Now dividing 255 by 102

255=102×2+51

Now dividing 102 by 51

102=51×2+0

So, 51 will by HCF.

ii)225 = 135 x 1 + 90

Since the remainder ≠ 0. So we apply the division lemma to the divisor 135 and remainder 90.

⇒ 135 = 90 x 1 + 45

Now we apply the division lemma to the new divisor 90 and remainder 45.

⇒ 90 = 45 x 2 + 0

Since the remainder at this stage is 0, the divisor will be the HCF.

Hence, the H.C.F of 225 and 135 is 45.

iii)616 = 32 × 19 + 8, since, here  r = 8  and  p ≠ 0 , thus, by continuing the process, we get

32 = 8 × 2 + 0 , here  r = 0

Thus, the HCF of 616 and 32 is equal to 8

Thus, the required maximum number of column = 8

   iv) HCF of 445 and 42 is 7.

Euclid's Division Lemma states that, if two positive integers “a” and “b”, then there exists unique integers “q” and “r” such that which satisfies the condition a = bq + r where 0 ≤ r ≤ b. ... In this example, 9 is the divisor, 58 is the dividend, 6 is the quotient and 4 is the remainder.

Euclid division algorithm states that Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

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