Power Mad
Work out 21, 22, 23, 24…
There are no powers of 2 that will be a multiple of 10 because the pattern keeps repeating itself. The pattern is that the first one ends in a 2, the next one ends in a 4, the next one ends in a 8, and the last one ends in a 6. This pattern happens because 2 doubled is 4, 4 doubled is 8, 8 doubled is 16 (which ends in a 6) , and 6 doubled is 12 (which ends in a 2). As 2, 4,8, and 6 doubled don’t end in a 0, no power of 2 will be a multiple of 10.
Work out 2n+3n for some different odd values of n.
I notice that it will always end in a 5 because 2x2x2=8, and 8x2x2=32, so for the odd powers of 2, it will end in 2, 8, 2, 8, and so on. For the odd powers of 3, it will end in 3, 7, 3, 7 and so on as 3x3x3=27, which ends in a 7, and 7x3x3=63, which ends in a three.
For which values of n is 1n+2n+3n is even?
When n is an integer which is 1 or more, the answer must be even as 1 to the power of anything must be 1, 2 to the power of anything will always be even, and 3 to the power of anything will always be odd. Two odds make an even, and two evens make an even (1+3=4, 4+2=6 – which is even), so the answer must always be even.
Work out 1n+2n+3n+4n for some different values of n.
I noticed that to the power of 1, 2, 3, will be a multiple of 10, but 4 won’t be a multiple of 10. The same for 5, 6, 7, but not 8. This pattern carries on in the sequence with 9, 10, 11, working, but not 12.
What about 1n+2n+3n+4n+5n?
When I worked out this question, I realised, it had a lot in common with the question before as 1, 2, and 3 all end in a 5 (a multiple of 5), however, 4 didn’t – it ended in a 9.When I looked more closely, I realised that the power of 5 to anything, so you just added a 5 to the units, which made it a multiple of 5 instead of 10. 4, 8, 12, and so on ended in a 9 as they used to end in a 4, but as we had to add 5 to the units, it became a 9, not a 4.
What other surprising results can you find?
When I was looking at adding and subtracting powers with each other, I found that there was always a pattern with the last numbers (usually with the same four numbers being the unit’s digit). When I was working them out, I was using Microsoft Excel to make it quicker. On the spread sheet, unfortunately, once the answer was over 15 digits, it was rounded the answer, which then meant that the pattern couldn’t be seen any more. The pattern was every four because all of the individual numbers to the power of something went round in a power of four, which therefore meant that when you added some of them together, it would still be a pattern of four.