Answer to Question #268728 in Algebra for dev

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

The trouble is the divisibility by the first 12 prime numbers,

so it must be a multiple of 2,3,5,7,11,13,17,19,23,29,31,37

To be odd it must look like 2K+1

to be a square it must look like$(2K+1)^2$ , and it must also be a cube

it must contain $(2K+1)^6$

so, it must have the form:

$2\times3\times5\times7\times11\times13\times17\times19\times23\times29\times31\times37(2K+1)^6$

when K = 0, we get

$2\times3\times5\times7\times11\times13\times17\times19\times23\times29\times31\times37(1)^6$

= $7.420738135\times 10^{12}$

which would be 13 digits long