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Shifting Times Tables

Stage: 3 Challenge Level: Challenge Level:1
Benita and Chloe, from Drumbowie, suggested the following method:

We found the differences between the numbers and looked at that times table, and then we found out if the table we had was shifted up or down from the times table it was meant to be, and how much by.

Nikita commented how much easier Levels 1 and 2 were because you simply need to find the nth term rule.

Jamie gave an example:

14, 24, 34, 44, 54
If the unit digits are identical, the table will be a multiple of ten and the shifted up number will be the same as the unit digit - here, 4.

He also gave some rules for helping to determine the times table:
  • If the numbers are all odd, the original table will be even and the shift will be odd.
  • If the numbers are all even, both the table and the shift will be even.
  • If the numbers are a mixture of odd and even, the table will be odd and the shift can be odd or even.
  • If there are only two different unit digits then the table is probably a multiple of five.
  • If the difference between two numbers is prime, then the table will be that prime number.
Lucia and Matt  gave similar rules. Matt went on to share a harder example:

348, 92, 252, 284, 124

The differences are even, so we can see that the times table is even. We can also see that this sequence has more than 2 different unit digits, so cannot be a multiple of either 5 or 10.

To work out exactly what the times table is we need to start by subtracting the lowest number from the second lowest number: 124 - 92 = 32. From this we can tell that the times table must be a factor of 32. We then need to subtract the second lowest number from the third lowest number (252 - 124 = 128), and then the next pair (284 - 252 = 32), and then the final pair (348 -284 = 64).

This gives us four numbers: 32,128,32 and 64. The highest common factor of these numbers is 32. From this we can see that the times table is 32 as all the numbers are a multiple of 32 apart.

To find the shift up or down we can divide one of the numbers in our times table and look at the remainder, e.g. 92/32 = 2 remainder 28. From this, we can see that if we were to move our number down 28 it would be a multiple of 32. This gives us our solution - the times table is 32 and you need to move it down 28.

 Well done to Sharanya who used the same method as Matt, and to everyone else who found a method.