HCF × LCM = Product of numbers. 36 × LCM = 408 × 468. LCM = (408 × 468) ÷ 36 = 190944 ÷ 36 = 5304.

Numbers are 10, 15, 20. 10 = 2 × 5, 15 = 3 × 5, 20 = 2² × 5. LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60 × 10 = 600.

Required number divides (247-7), (328-7), (409-7) = 240, 321, 402. HCF(240, 321, 402) = 120.

HCF × LCM = Product of numbers = 360. 360 = 20 × other number.

Other number = 360 ÷ 20 = 18.

They ring together after LCM(12, 18) minutes. 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 36 minutes.

144 = 2⁴ × 3², 180 = 2² × 3² × 5, 192 = 2⁶ × 3.

Common factors: 2² and 3. HCF = 2² × 3 = 12.

Using HCF × LCM = Product: 6 × 60 = 12 × other. 360 = 12 × other.

Other = 360 ÷ 12 = 30.

Numbers are 7, 14, 21, 28. 7 = 7, 14 = 2 × 7, 21 = 3 × 7, 28 = 2² × 7. LCM = 2² × 3 × 7 = 4 × 3 × 7 × 2 = 1680.

Using the formula: HCF × LCM = Product of two numbers

12 × 180 = 60 × Other number

2160 = 60 × Other number

Other number = 2160 ÷ 60 = 36

Verification: HCF(60, 36) = 12 ✓ and LCM(60, 36) = 180 ✓

Using HCF × LCM = Product of two numbers:

15 × 450 = 90 × Other number

6750 = 90 × Other number

Other number = 6750 ÷ 90 = 75

Verification: HCF(90, 75) = 15 ✓ and LCM(90, 75) = 450 ✓