Govt Exams
They ring together after LCM(12, 18) minutes. 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 36 minutes.
HCF × LCM = Product of numbers. 36 × LCM = 408 × 468. LCM = (408 × 468) ÷ 36 = 190944 ÷ 36 = 5304.
12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. LCM = 2³ × 3² = 8 × 9 = 72.
Let numbers be 3k and 5k.
Their HCF is k = 8.
So numbers are 3(8) = 24 and 5(8) = 40.
LCM(12, 18) = 36.
Time for both = 36/(36/12 + 36/18) = 36/(3 + 2) = 36/5 = 7.2 minutes.
150 = 2 × 3 × 5², 200 = 2³ × 5².
Common factors: 2 and 5². HCF = 2 × 5² = 2 × 25 = 50.
HCF × LCM = Product of numbers. 12 × 144 = 36 × other. 1728 = 36 × other.
Other = 1728 ÷ 36 = 48.
Using HCF × LCM = Product of numbers: 12 × 72 = 24 × other number. 864 = 24 × other number.
Other number = 864 ÷ 24 = 36.
Using HCF × LCM = Product of numbers: 4 × 60 = 20 × other.
Therefore, other = 240 ÷ 20 = 12.
110 = 2 × 5 × 11, 165 = 3 × 5 × 11, 220 = 2² × 5 × 11.
Common factors: 5 × 11 = 55.