Compute the positive integer less than 1000 which has exactly 29 positive proper divisors.
A number or a short expression. Spacing and $ signs are ignored.
Solution
Recall that the number N=p1e1p2e2⋯pkek (where the pi are distinct primes) has exactly (e1+1)(e2+1)⋯(ek+1) positive integer divisors including itself. We seek N<1000 such that this expression is 30. Since 30=2⋅3⋅5, we take e1=1,e2=2,e3=4. Then we see that N=513242=720 is satisfactory.
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Source: Omni-MATH,
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