if A is greater than B B is greater than CC is greater than D, D is greater than is e are consecutive positive integers such that b + C + D is a perfect square A + B + C + b + c is the perfect to the smallest possible value is
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Since the middle term of an arithmetic progression with an odd number of terms is the average of the series,
we know b+c+d=3c and a+b+c+d+e=5c.
Thus, c must be in the form of 3⋅x
2
based upon the first part
and in the form of 5
2
⋅y
3
based upon the second part, with x and y denoting an integers.
c is minimized if it’s prime factorization contains only 3,5,
and since there is a cubed term in 5
2
⋅y
3
, 3
3
must be a factor of c.
3
3
5
2
=675
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