Work out 2n+3n for some different odd values of n.
What do you notice?
2n
1 2
3 8
5 32
2 and 8
2 and 8 are even because 2n equations always end in an even number and you can’t multiply a number by 2 and end up with an odd number.
The numbers are 2 and 8 because 2=2, 2x2=4, 2x2x2=8, 2x2x2x2=16, 2x2x2x2x2=32. This is shown as 2=2, 2x4=8, 8x4= 32 2x4=8 and so on. This shows that all the alternate ns in 2n will always end with a 2 or an 8.
3n
1 3
3 27
5 243
3 and 7
3 and 5 are odd as if you multiply an odd number by another odd number it always ends up as an odd number. This means that even if a number had 7 different digits before an odd number at the end and you multiplied it by another odd number it will still always end up as an odd number.
I have proven that 3n will always end in an odd number as 3=3, 3x3=9, 3x3x3=27, 3x3x3x3=81, 3x3x3x3x3=243. This is shown as 3=3, 3x9=27, 7x9=63, 3x9=27 and the pattern starts again. This shows that alternate ns in 3n will always end in a 3 or a 5.
Also the units cannot be affected by the tens as if you double a ten e.g. 60 it will just double itself and wouldn’t affect the units, 120.
2n+3n
1+1 5
3+3 35
5+5 275
5
The numbers always end in 5 as 2+3 is 5 and 8+7 is 15 which ends in a 5 so it always ends in a 5.
So in conclusion all the odd n numbers like 23+33 end in 5 as 2+3 and 8+7 always end in 5.