Define HCF of two positive integers and find the HCF of the following pairs of numbers:
Question:

Define HCF of two positive integers and find the HCF of the following pairs of numbers:

(i) 32 and 54

(ii) 18 and 24

(iii) 70 and 30

(iv) 56 and 88

(v) 475 and 495

(vi) 75 and 243.

(vii) 240 and 6552

(viii) 155 and 1385

(ix) 100 and 190

(x) 105 and 120

Solution:

(i) We need to find H.C.F. of 32 and 54.

By applying division lemma

$54=32 \times 1+22$

Since remainder $\neq 0$, apply division lemma on 32 and remainder 22

$32=22 \times 1+10$

Since remainder $\neq 0$, apply division lemma on 22 and remainder 10

$22=10 \times 2+2$

Since remainder $\neq 0$, apply division lemma on 10 and remainder 2

$10=2 \times 5+0$

Therefore, H.C.F. of 32 and 54 is 2

(ii) We need to find H.C.F. of 18 and 24.

By applying division lemma

$24=18 \times 1+6$

Since remainder $\neq 0$, apply division lemma on divisor 18 and remainder 6

$18=6 \times 3+0$

Therefore, H.C.F. of 18 and 24 is 6

(iii) We need to find H.C.F. of 70 and 30.

By applying Euclid’s Division lemma

$70=30 \times 2+10$

Since remainder $\neq 0$, apply division lemma on divisor 30 and remainder 10

$30=10 \times 3+0 .$

Therefore, H.C.F. of 70 and $30=10$

(iv) We need to find H.C.F. of 56 and 88.

By applying Euclid’s Division lemma

$88=56 \times 1+32$

Since remainder $\neq 0$, apply division lemma on 56 and remainder 32

$56=32 \times 1+24$

Since remainder $\neq 0$, apply division lemma on 32 and remainder 24

$32=24 \times 1+8 .$

Since remainder $\neq 0$, apply division lemma on 24 and remainder 8

$24=8 \times 3+0$

Therefore, H.C.F. of 56 and $88=8$.

(v) We need to find H.C.F. of 475 and 495.

By applying Euclid’s Division lemma

$495=475 \times 1+20$

Since remainder $\neq 0$, apply division lemma on 475 and remainder 20

$475=20 \times 23+15$

Since remainder $\neq 0$, apply division lemma on 20 and remainder 15

$20=15 \times 1+5$

Since remainder $\neq 0$, apply division lemma on 15 and remainder 5

$15=5 \times 3+0$

Therefore, H.C.F. of 475 and $495=5$.

(vi) We need to find H.C.F. of 75 and 243.

By applying Euclid’s Division lemma

$243=75 \times 3+18$

Since remainder $\neq 0$, apply division lemma on 75 and remainder 18

$75=18 \times 4+3$

Since remainder $\neq 0$, apply division lemma on divisor 18 and remainder 3

$18=3 \times 6+0$

Therefore, H.C.F. of 75 and $243=3$.

(vii) We need to find H.C.F. of 240 and 6552.

By applying Euclid’s Division lemma

$6552=240 \times 27+72$

Since remainder $\neq 0$, apply division lemma on divisor 240 and remainder 72

$240=72 \times 3+24$

Since remainder $\neq 0$, apply division lemma on divisor 72 and remainder 24

$72=24 \times 3+0$

Therefore, H.C.F. of 240 and $6552=24$.

(viii) We need to find H.C.F. of 155 and 1385.

By applying Euclid’s Division lemma 

$1385=155 \times 8+145$

Since remainder $\neq 0$, apply division lemma on divisor 155 and remainder 145

$155=145 \times 1+10$

Since remainder $\neq 0$, apply division lemma on divisor 145 and remainder 10

$145=10 \times 14+5$

Since remainder $\neq 0$, apply division lemma on divisor 10 and remainder 5

$10=5 \times 2+0 .$

Therefore, H.C.F. of 155 and $1385=5$.

(ix) We need to find H.C.F. of 100 and 190.

By applying Euclid’s division lemma

$190=100 \times 1+90$

Since remainder $\neq 0$, apply division lemma on divisor 100 and remainder 90

$100=90 \times 1+10 .$

Since remainder $\neq 0$, apply division lemma on divisor 90 and remainder 10

$90=\times 10 \times 9+0$

Therefore, H.C.F. of 100 and $190=10$.

(x) We need to find H.C.F. of 105 and 120.

By applying Euclid’s division lemma 

$120=105 \times 1+15$

Since remainder $\neq 0$, apply division lemma on divisor 105 and remainder 15

$105=15 \times 7+0$

Therefore, H.C.F. of 105 and $_{120}=15$.

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