Answer to Question #268390 in Combinatorics | Number Theory for Naman

Question

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?

Answer 1

Let;

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

$\therefore$ 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)

$\therefore N = 7420738134810^6 = 166986990622241277447975946146208311466303689437571218130959563879576881000000$