.
a five digit base ten number is the fourth power of an integer. The sum of the first, third,
and fifth digits equals the sum of the second and fourth digits. What digit is in the thousands place?
~~~~~~~~~~~~~~~~
It is a nice problem to apply the divisibility rules.
Since the sum of the first, third and fifth digits equals the sum of the second and fourth digits,
it means that the alternate sum (or difference) of the digits is 0 (zero, ZERO).
Then the rule of divisibility by 11 tells us that this 5-digit number is divisible by 11.
Since 11 is a prime number, it implies that the 5-digit number is a multiple of 11^4, which is 14641.
Among 5-digit numbers, there is ONLY ONE such number (4-th degree of a multiple of 11),
since the next such number 22^4 = 234256 is just 6-digit number.
So, only one number satisfies the problem's condition, and it is the number 14641.
Its thousands digit is 4, which gives the answer to the problem's question.
Solved.
==============
On the divisibility by 11 rule, see the lesson
- Divisibility by 11 rule
in this site.