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Thanks to the group of pupils from River Valley High School, Singapore, (Chong Ching Tong, Chen Wei Jian and Teo Seow Tian). They demonstrated that the rule worked for a range of numbers and identified some patterns; namely that 70, 21 and 15 are multiples of combinations of two of the divisors (3, 5 and 7) and that 105 is the lowest common multiple of 3, 5 and 7. They looked at combinations of multiples of 3, 5 and 7, some of which are given below. However, they were not quite able to generalise what they had discovered so I think there is a little more work to do on this. I have given you a hint for the next step at the end.
After testing out a few times, we managed to explain how it works for multiples of 3:
18 ÷ 3 = 6 Remainder = 0 | 0 * 70 = 0 |
18 ÷ 5 = 3 Remainder = 3 | 3 * 21 = 63 |
18 ÷ 7 = 2 Remainder = 4 | 4 * 15 = 60 |
After testing out a few times, we managed to explain how it works for multiples of 5:
20 ÷ 3 = 6 Remainder = 2 | 2 * 70 = 140 |
20 ÷ 5 = 4 Remainder = 0 | 0 * 21 = 21 |
20 ÷ 7 = 2 Remainder = 6 | 6 * 15 = 90 |
After testing out a few times, we managed to explain how it works for multiples of 3 and 5:
60 ÷ 3 = 20 Remainder = 0 | 0 * 70 = 0 |
60 ÷ 5 = 12 Remainder = 0 | 0 * 21 = 0 |
60 ÷ 7 = 8 Remainder = 4 | 4 * 15 = 60 |
After testing out a few times, we managed to explain how it works for prime numbers:
13 ÷ 3 = 4 Remainder = 1 | 1 * 70 = 70 |
13 ÷ 5 = 2 Remainder = 3 | 3 * 21 = 63 |
13 ÷ 7 = 1 Remainder = 6 | 6 * 15 = 90 |
Hint
A possible route to the solution might be to use the idea that when one number (n say) is divided by another number (d) the answer can be written as a whole number (q) with a remainder (r).
That is n/d = q Remainder r
This can also be thought of as: n = qd + r
So, for example, 32/5 = 6 Remainder 2
This can be thought of as : 32 = 6 x 5 + 2