Question #268390

Find the smallest positive integer N that satisfies all of the following conditions:


• N is a square.


• N is a cube.


• N is an odd number.


• N is divisible by twelve prime numbers.


How many digits does this number N have?


1
Expert's answer
2021-11-21T15:56:04-0500

Let;

x2=N;y3=NN=oddN2×3×5×7×11×13×17×19×23×29×31×37=z (where z is an integer)N=7420738134810zx² = N;\\ y³ = N\\ N = \textsf{odd}\\ \dfrac{N}{2×3×5×7×11×13×17×19×23×29×31×37}= z \ (\textsf{where z is an integer})\\ N=7420738134810z


\therefore N is a multiple of 7420738134810


for N to be a square and a cube, and to be a multiple of a multiple of prime numbers, the multiple of prime numbers have to be exponentiated to (2×3)

N=74207381348106=166986990622241277447975946146208311466303689437571218130959563879576881000000\therefore N = 7420738134810^6 = 166986990622241277447975946146208311466303689437571218130959563879576881000000

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