MAT question

How do we do this? Is there a property of square numbers that would help?

Exactly one of these five numbers is a square number. Which one?
(a) 99,999,999, (b) 123,333,333, (c) 649,485,225, (d) 713,291,035, (e) 987,654,000.

How do we do this? Is there a property of square numbers that would help?

Exactly one of these five numbers is a square number. Which one?
(a) 99,999,999, (b) 123,333,333, (c) 649,485,225, (d) 713,291,035, (e) 987,654,000.

a) is 1 off 100,000,000 which is a square number (even number of zeros), so that cant be a square
b) divisble by 3 but not 9 (digit sum) so not square
d) has 5^1 in prime factorisation (not a multiple of 25) so not square
e) has 5^3 in the prime factorisation so not square

so c)

I see, so I can also deduce info from its prime factorisation. So generalising d and e, if a number has a prime factor that is a square, the number is also a square?

As a start, what are the possible last digits for a square number?
You might also want to think about the highest power of 5 dividing each number, and if that's compatible with it being square.

You should probably know this, but a square number has prime factors with even exponents. Otherwise when you root it (half the exponents), the result will not be an integer.