If 2^P and 5^P start with the same (non-zero) digit for positive integer P, what is that digit? Can you prove it must be so?

Write down the squares of the integers from 1 to 9:

1:  1
2:  4
3:  9
4: 16
5: 25
6: 36
7: 49
8: 64
9: 81


Therefore the product of two numbers with the same starting digit i has its first digit in the following range:

1: 1-3
2: 4-9
3: 9 or 1
4: 1-2
5: 2-3
6: 3-4
7: 4-6
8: 6-8
9: 8-9


The product of 2^p and 5^p is 10^p and starts with 1, therefore the starting digit of 2^p and 5^p could only be 1, 3 or 4.

1 does not work because the product of 2^p and 5^p would have non-zero's somewhere after the starting digit, which is only possible if 2^p and 5^p have zero's only, which is not the case.

4 does not work because the second digit would be between 6-9.

Leaves only 3.

Posted by JLo on 2006-12-03 12:13:40