Old but interesting problem

Number Theory Level 2

$\large \left \lfloor \dfrac{3^{100}}{100} \right \rfloor + 10^{50}$

Is the above expression divisible by 3?

Note : This question can be solved without using a calculator.

Yes No

1 solution

Mike Vanderheyden
Mar 26, 2021

The result of the expression will be divisible by 3 if the sum of its digits is 0 mod 3.
The sum of the digits of $3^{100}$ is 0 mod 3.
10 to any power will result in a 1 followed by some number of zeroes, so the sum of the digits of $10^{50}$ is 1.
Dividing an integer by 100 and then taking the floor is akin to simply removing the two rightmost digits.

So all that remains is to figure out the last two digits of $3^{100}$ .

The units digit of powers of 3 will cycle through (3, 9, 7, 1) so the 50th of these will be 9.

The tens digit will cycle through (0, 0, 2, 8, 4, 2, 8, 6, 8, 4, 4, 4, 2, 6, 0, 2, 6, 8, 6, 0). The 50th of these will be 4.

So putting it all together we have [some big number whose sum of digits is 0 mod 3] - 9 - 4 + 1

= [some big number whose sum of digits is 0 mod 3] - 12

= [some other big number whose sum of digits is still 0 mod 3].