Q1.
Find the common factors and the HCF of the following numbers: (a) 50, 60 (b) 140, 275 (c) 77, 725 (d) 370, 592 (e) 81, 243
Answer
Factorise both numbers, keep the shared primes, then list every subpart built from them.
(a) 50 = 2 × 5 × 5, 60 = 2 × 2 × 3 × 5 → shared: one 2, one 5 → HCF = 2 × 5 = 10
(b) 140 = 2 × 2 × 5 × 7, 275 = 5 × 5 × 11 → shared: one 5 → HCF = 5
(c) 77 = 7 × 11, 725 = 5 × 5 × 29 → shared: none → HCF = 1
(d) 370 = 2 × 5 × 37, 592 = 2 × 2 × 2 × 2 × 37 → shared: one 2, one 37 → HCF = 2 × 37 = 74
(e) 81 = 3 × 3 × 3 × 3, 243 = 3 × 3 × 3 × 3 × 3 → shared: four 3s → HCF = 3 × 3 × 3 × 3 = 81
(b) 140 = 2 × 2 × 5 × 7, 275 = 5 × 5 × 11 → shared: one 5 → HCF = 5
(c) 77 = 7 × 11, 725 = 5 × 5 × 29 → shared: none → HCF = 1
(d) 370 = 2 × 5 × 37, 592 = 2 × 2 × 2 × 2 × 37 → shared: one 2, one 37 → HCF = 2 × 37 = 74
(e) 81 = 3 × 3 × 3 × 3, 243 = 3 × 3 × 3 × 3 × 3 → shared: four 3s → HCF = 3 × 3 × 3 × 3 = 81
| Numbers | Common factors | HCF | |
|---|---|---|---|
| (a) | 50, 60 | 1, 2, 5, 10 | 10 |
| (b) | 140, 275 | 1, 5 | 5 |
| (c) | 77, 725 | 1 | 1 |
| (d) | 370, 592 | 1, 2, 37, 74 | 74 |
| (e) | 81, 243 | 1, 3, 9, 27, 81 | 81 |
Why it happens: the common factors are exactly the factors of the HCF. That is why each row's list is just the factor list of the number in the last column — 1, 2, 5, 10 are the factors of 10; 1, 3, 9, 27, 81 are the factors of 81. Once you know the HCF, you know all the common factors for free.
Tip: in (e), 81 is itself a factor of 243 (243 = 81 × 3). Whenever one number divides the other, the smaller one is the HCF.