Find the L.C.M. and H.C.F. of the following pairs of integers and verify that L.C.M. × H.C.F. = product of two numbers.

(i) 26 and 91

(ii) 510 and 92

(iii) 336 and 54

We know that,

H.C.F. = Product of smallest power of each common prime factor in the numbers.

L.C.M. = Product of the greatest power of each prime factor, involved in the numbers.

(i) On prime factorizing, we get :

26
= 2 × 13

91
= 7 × 13

H.C.F. (26, 91) = 13

L.C.M. (26, 91) = 2 × 7 × 13 = 182.

Product of numbers = 26 × 91 = 2366

Product of H.C.F. and L.C.M. = 13 × 182 = 2366.

∴ Product of the two numbers = Product of H.C.F. and L.C.M.

Hence, H.C.F. = 13 and L.C.M. = 182.

(ii) On prime factorizing, we get :

510
= 2 × 255
= 2 × 3 × 85
= 2 × 3 × 5 × 17

92
= 2 × 46
= 2 × 2 × 23
= 2² × 23

H.C.F. (510, 92) = 2

L.C.M. (510, 92) = 2² × 3 × 5 × 17 × 23 = 23460.

Product of numbers = 510 × 92 = 46920

Product of H.C.F. and L.C.M. = 2 × 23460 = 46920.

∴ Product of the two numbers = Product of H.C.F. and L.C.M.

Hence, H.C.F. = 2 and L.C.M. = 23460.

(iii) On prime factorizing, we get :

336
= 2 × 168
= 2 × 2 × 84
= 2 × 2 × 2 × 42
= 2 × 2 × 2 × 2 × 21
= 2 × 2 × 2 × 2 × 3 × 7
= 2⁴ × 3 × 7

54
= 2 × 27
= 2 × 3 × 9
= 2 × 3 × 3 × 3
= 3³ × 2

H.C.F. (336, 54) = 2 × 3 = 6

L.C.M. (336, 54) = 2⁴ × 3³ × 7 = 3024.

Product of numbers = 336 × 54 = 18144

Product of H.C.F. and L.C.M. = 6 × 3024 = 18144.

∴ Product of numbers = Product of H.C.F. and L.C.M.

Hence, H.C.F. = 6 and L.C.M. = 3024.