Question

For each of the following pairs of numbers, verify the property:
Product of the number = Product of their H.C.F and L.C.M

(i) 25,65
(ii) 117,221
(iii) 35,40
(iv) 87,145
(v) 490,1155

Answer 1

(i) Given numbers are 25 and 65.

Prime factorisation of 25 = 5 × 5
Prime factorisation of 65 = 5 × 13

HCF of 25 and 65 = 5
LCM of 25 and 65 = 5 × 5 × 13 = 325
Product of the given numbers = 25 × 65 = 1,625
Product of their HCF and LCM = 5 × 325 = 1,625

∴ Product of the number = Product of their HCF and LCM (Verified)

(ii) Given numbers are 117 and 221.

Prime factorisation of 117 = 3 × 3 × 13
Prime factorisation of 221 = 13 × 17

HCF of 117 and 221 = 13
LCM of 117 and 221 = 3 × 3 × 13 × 17 = 1,989
Product of the given number = 117 × 221 = 12,857
Product of their HCF and LCM = 13 × 1,989 = 12,857

∴ Product of the number = Product of their HCF and LCM (Verified)

(iii) Given numbers are 35 and 40.

Prime factorisation of 35 = 5 × 7
Prime factorisation of 40 = 2 × 2 × 2 × 5

HCF of 35 and 40 = 5
LCM of 35 and 40 = 2 × 2 × 2 × 5 × 7 = 280
Product of the given number = 35 × 40 = 1,400
Product of their HCF and LCM = 5 × 280 = 1,400

∴ Product of the number = Product of their HCF and LCM (Verified)

(iv) Given numbers are 87 and 145.

Prime factorisation of 87 = 3 × 29
Prime factorisation of 145 = 5 × 29

HCF of 87 and 145 = 29
LCM of 87 and 145 = 3 × 5 × 29 = 435
Product of the given number = 87 × 145 = 12,615
Product of their HCF and LCM = 29 × 435 = 12,615

∴ Product of the number = Product of their HCF and LCM (Verified)

(v) Given numbers are 490 and 1,155.

Prime factorisation of 490 = 2 × 5 × 7 × 7
Prime factorisation of 1,155 = 3 × 5 × 7 × 11

HCF of 490 and 1,155 = 5 × 7 = 35
LCM of 490 and 1,155 = 2 × 3 × 3 × 5 × 7 × 7 × 11 = 16,170
Product of the given number = 490 × 1,155 = 5,65,950
Product of their HCF and LCM = 35 × 16,170 = 5,65,950

∴ Product of the number = Product of their HCF and LCM (Verified)