find the H.C.F. of the following pairs of numbers and also express it as a linear combination of them
(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
Answers
Step-by-step explanation:(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(i×) 100 and 190(×) 105 and 120Solution:(i) We need to find H.C.F. of 32 and 54.By applying division lemma 54 = 32 × 1 + 22Since remainder ≠ 0, apply division lemma on 32 and remainder 2232 = 22 × 1 + 10Since remainder ≠ 0, apply division lemma on 22 and remainder 1022 = 10 × 2 + 2Since remainder ≠ 0, apply division lemma on 10 and 210 = 2 × 5 + 0Therefore, H.C.F. of 32 and 54 is(ii) We need to find H.C.F. of 18 and 24.By applying division lemma24 = 18 × 1 + 6.Since remainder ≠ 0, apply division lemma on divisor 18 and remainder 618 = 6 × 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 lemma70 = 30 × 2 + 10.Since remainder ≠ 0, apply division lemma on divisor 30 and remainder 1030 = 10 × 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 lemma88 = 56 × 1 + 32.Since remainder ≠ 0, apply division lemma on 56 and remainder 3256 = 32 × 1 + 24.Since remainder ≠ 0, apply division lemma on 32 and remainder 2432 = 24 × 1+ 8.Since remainder ≠ 0, apply division lemma on 24 and remainder 824 = 8 × 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 × 1 + 20.Since remainder ≠ 0, apply division lemma on 475 and remainder 20475 = 20 × 23 + 15.Since remainder ≠ 0, apply division lemma on 20 and remainder 1520 = 15 × 1 + 5.Since remainder ≠ 0, apply division lemma on 15 and remainder 515 = 5 × 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 lemma243 = 75 × 3 + 18.Since remainder ≠ 0, apply division lemma on 75 and remainder 1875 = 18 × 4 + 3.Since remainder ≠ 0, apply division lemma on divisor 18 and remainder 318 = 3 × 6 + 0.We need to find H.C.F. of 155 and 1385.By applying Euclid’s Division lemma1385 = 155 × 8 + 145.Since remainder ≠ 0, apply division lemma on divisor 155 and remainder 145.155 = 145 × 1 + 10.Since remainder ≠ 0 apply division lemma on divisor 145 and remainder 10145 = 10 × 14 + 5.Since remainder ≠ 0, apply division lemma on divisor 10 and remainder 510 = 5 × 2 + 0.Therefore, H.C.F. of 155 and 1385 = 5(i×) We need to find H.C.F. of 100 and 190.By applying Euclid’s division lemma190 = 100 × 1 + 90.Since remainder ≠ 0, apply division lemma on divisor 100 and remainder 90100 = 90 × 1 + 10.Since remainder ≠ 0, apply division lemma on divisor 90 and remainder 1090 = 10 × 9 + 0.Therefore, H.C.F. of 100 and 190 = 10(×) We need to find H.C.F. of 105 and 120.By applying Euclid’s division lemma120 = 105 × 1 + 15.Since remainder ≠ 0, apply division lemma on divisor 105 and remainder 15105 = 15 × 7 + 0.Therefore, H.C.F. of 105 and 120 = 15.
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