Math word problem!! how old is....? please?

Question

the children's ages combined is three times the age of mike, who is three years younger than Shawn. The cube [^3] of Shawn's age, plus the square [^2] of Mikes age plus Laura's age totals a multiple of 100. IF ALL THE AGES ARE IN YEARS, HOW OLD IS LAURA?
(also tell the ages of the other children anyway)
showing work would help u get closer to the ten points but make sure its readable! PLEASE??

Answer 1

There's an error in what's below, but it may be helpful anyway as a source of ideas.

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L + M + S = 3M

S = 3 + M

So L = M -3 (substituting)

S^3 + M^2 + L is divisible by 100.

Hmm. Let's do this in terms of M.

(M+3)^3 + M^2 + M -3 is divisible by 100.

That's M^3 + 4M^2 +4M + 24.

So M has to be even. Call it 2N

2N^3 +4N^2 + 2N + 6 is divisible by 25 (dividing out a factor of 4). Hence so is N^3 + 2N ^2 + N + 3 (pulling out a factor of 2, since 2 and 25 are relatively prime).

That has no rational roots, so factoring it is unlikely to help.

Well, let's look at it modulo 5. N can't be congruent to 0. Or to 1. Or to 2. Or to 3. Or to 4.

Sorry. I obviously made an error.

But generally this seems like a good approach.