Q1.
Do you remember the ‘Jump Jackpot’ game from Grade 6 (see the chapter ‘Prime Time’)? Grumpy places a treasure on a number and Jumpy chooses a jump size and tries to collect the treasure. In each case below, the two numbers upon which treasures are kept are given. Find the longest jump size (starting from 0) using which Jumpy can land on both the numbers having the treasure. (a) 14 and 30 (b) 7 and 11 (c) 30 and 50 (d) 28 and 42
Answer
Jumping from 0 in steps of size j, Jumpy lands on 0, j, 2j, 3j, … — the multiples of j. To land on a treasure number, j must be a factor of it.
| Numbers | Common factors | Longest jump | |
|---|---|---|---|
| (a) | 14 and 30 | 1, 2 | 2 |
| (b) | 7 and 11 | 1 | 1 |
| (c) | 30 and 50 | 1, 2, 5, 10 | 10 |
| (d) | 28 and 42 | 1, 2, 7, 14 | 14 |
(a) 14 = 2 × 7, 30 = 2 × 3 × 5 → only 2 is shared → jump 2
(b) 7 and 11 are both prime and different → only 1 is shared → jump 1
(c) 30 = 2 × 3 × 5, 50 = 2 × 5 × 5 → 2 and 5 shared → jump 2 × 5 = 10
(d) 28 = 2 × 2 × 7, 42 = 2 × 3 × 7 → 2 and 7 shared → jump 2 × 7 = 14
(b) 7 and 11 are both prime and different → only 1 is shared → jump 1
(c) 30 = 2 × 3 × 5, 50 = 2 × 5 × 5 → 2 and 5 shared → jump 2 × 5 = 10
(d) 28 = 2 × 2 × 7, 42 = 2 × 3 × 7 → 2 and 7 shared → jump 2 × 7 = 14
Check it yourself: for (c), jumps of 10 land on 0, 10, 20, 30, 40, 50 — both treasures collected. A jump of 15 lands on 30 but skips over 50.