Question

how many factors of 4000 are perfect squares

Answer 1

start by working its prime factorization

Answer 2

and see how many ways you can choose different combinations of `powers of squares of prime numbers`

Answer 3

\[\large 4000 = 4\times 1000 = 2^5\times 5^3 = 2\cdot (2^2)^2\times 5\cdot5^2 = 2.a^2\times 5\cdot b\]

Answer 4

\[\large a^2 \cdot b^1\]
How many ways can you choose exponents of \(\large a\) and \(\large b\) ?

Answer 5

exponent of \(\large a\) can be 0,1 or 2 : 3 ways
exponent of \(\large b\) can be 0 or 1 : 2 ways

multiply them to get the total number of perfect square divisors : 3x2 = 6

Answer 6

Note that that count includes the trivial perfect square divisor : 1