What's the second smallest integer with a units digit of 5, which is the square of an integer, and the cube of an integer?
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Let's call the number we are looking for n.
we know it's the square of an integer, so:
n=int2a
and we know it's the cube of an integer, so:
n=int3b
since any integer can be prime-factorized, let's consider the prime factorization of int1 and int2:
inta=primeinta11⋅primeinta22...
intb=primeintb11⋅primeintb22...
n=(inta)2=(primeinta11⋅primeinta22...)2=prime2⋅inta11⋅prime2⋅inta22...
and
n=int3b=(primeintb11⋅primeintb22...)3=prime3⋅intb11⋅prime3⋅intb22...
so prime2⋅inta11⋅prime2⋅inta22...=prime3⋅intb11⋅prime3⋅intb22...
and so we can say:
prime2⋅intaxx=prime3⋅intbxx for all x.
and therefore: 2⋅intax=3⋅intbx this is the number of primex in the prime factorization of n.
since it has a factor of 2 and 3, then the number of primex in n must be a multiple of 6.
to summarize, for any prime factor of n,there are a multiple of 6 of them in n.
we also know that the units digit of n is 5, which means it has a factor of 5, so:
n=5⋅int3
so n has at least one 5 in it's prime factorization, and consequently it must have at least 6.
so 56 is the least integer that fits the description, but we are looking for the 2nd smallest.
well 512 would fit the description, and it's not the smallest, but maybe it's not the 2nd smallest,
56⋅26 is worth considering, but 56⋅26=106, which is one million, which doesn't have a 5 as the units digit. This does not fit the description in the problem.
56⋅36 fits the description and is the second smallest integer that does.
n=56⋅36=11390625
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