Hint
Try starting with the cube root of the number.
Answer
195 x 197 x 199.
Reasoning
Since we're after three numbers that multiply together, a good place to start is the cube root of 7,644,585, which is roughly 196.99.
Let's try dividing by the closest odd number to 196.99:
7644585 ÷ 197 = 38805
We're now after whole divisors of 38,805, so trying either odd number either side of 197 might work.
Trying either 195 or 199 gives the answer.
Double-Checking
195 x 197 x 199 = 7,644,585
Note
Trying the closest odd number to the cube root always works, and the other two numbers are the odd numbers either side.
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Puzzle 2
How many animals can you find in this rather curious poem:
A person, so simple we are.
Catch the kid o'er the bridge.
Follow the chief, oxtail soup we like.
Anagram ANPI gives us PAIN, cower under a ridge.
My 6, 7, 3 is a tear to a newspaper.
My 5, 4, 1 is damper than moist.
My 1, 7, 6, 4 is to become weary.
My 5, 7, 6, 9 helps electricity to flow.
My 8, 10, 2 is an attempt at this puzzle.