Q1.
Find the LCM of the following numbers: (a) 30, 72 (b) 36, 54 (c) 105, 195, 65 (d) 222, 370
Answer
Take every prime that appears, each the maximum number of times.
| Prime factorisations | Maximum count of each prime | LCM | |
|---|---|---|---|
| (a) | 30 = 2 × 3 × 5 72 = 2 × 2 × 2 × 3 × 3 | three 2s, two 3s, one 5 | 2 × 2 × 2 × 3 × 3 × 5 = 360 |
| (b) | 36 = 2 × 2 × 3 × 3 54 = 2 × 3 × 3 × 3 | two 2s, three 3s | 2 × 2 × 3 × 3 × 3 = 108 |
| (c) | 105 = 3 × 5 × 7 195 = 3 × 5 × 13 65 = 5 × 13 | one 3, one 5, one 7, one 13 | 3 × 5 × 7 × 13 = 1365 |
| (d) | 222 = 2 × 3 × 37 370 = 2 × 5 × 37 | one 2, one 3, one 5, one 37 | 2 × 3 × 5 × 37 = 1110 |
Checks:
(a) 360 ÷ 30 = 12, 360 ÷ 72 = 5 ✓
(b) 108 ÷ 36 = 3, 108 ÷ 54 = 2 ✓
(c) 1365 ÷ 105 = 13, 1365 ÷ 195 = 7, 1365 ÷ 65 = 21 ✓
(d) 1110 ÷ 222 = 5, 1110 ÷ 370 = 3 ✓
(a) 360 ÷ 30 = 12, 360 ÷ 72 = 5 ✓
(b) 108 ÷ 36 = 3, 108 ÷ 54 = 2 ✓
(c) 1365 ÷ 105 = 13, 1365 ÷ 195 = 7, 1365 ÷ 65 = 21 ✓
(d) 1110 ÷ 222 = 5, 1110 ÷ 370 = 3 ✓
Tip: in (d) the LCM 1110 is much smaller than the product 222 × 370 = 82,140 — because the two numbers share 2 and 37. In fact HCF(222, 370) = 74, and 74 × 1110 = 82,140.