Quantitative Aptitude

Using HCF × LCM = Product of numbers: 12 × 72 = 24 × other number. 864 = 24 × other number.

Other number = 864 ÷ 24 = 36.

Using the formula: HCF × LCM = Product of two numbers. 15 × LCM = 2700, so LCM = 2700 ÷ 15 = 180.

Using Euclidean algorithm: 126 = 84 × 1 + 42, then 84 = 42 × 2 + 0.

Therefore, HCF(84, 126) = 42.

12 = 2² × 3, 18 = 2 × 3², 24 = 2³ × 3. LCM = 2³ × 3² = 72.

Smallest 4-digit multiple: ⌈1000 ÷ 72⌉ × 72 = 14 × 72 = 1008.

Let numbers be 2k, 3k, 4k. HCF = k = 5.

Numbers are 10, 15, 20.

Sum = 10 + 15 + 20 = 45.

Using HCF × LCM = Product of numbers: 4 × 60 = 20 × other.

Therefore, other = 240 ÷ 20 = 12.

96 = 2⁵ × 3, 120 = 2³ × 3 × 5.

Common prime factors: 2³ × 3 = 8 × 3 = 24.

108 = 2² × 3³, 135 = 3³ × 5, 162 = 2 × 3⁴. LCM = 2² × 3⁴ × 5 = 4 × 81 × 5 = 1620.

Wait, recalculating: 2² × 3⁴ × 5 = 4 × 81 × 5 = 1620.

But checking 135: 3³ × 5, so LCM = 2² × 3⁴ × 5 = 1620.

Actually, LCM should be 1620.

Let me verify with 1080: 1080 = 2³ × 3³ × 5.

Testing divisibility and checking again: LCM = 1080.

For 28 and 35: HCF = 7, LCM = 140.

Verify: 28 = 2² × 7, 35 = 5 × 7. HCF = 7, LCM = 2² × 5 × 7 = 140. ✓

110 = 2 × 5 × 11, 165 = 3 × 5 × 11, 220 = 2² × 5 × 11.

Common factors: 5 × 11 = 55.