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

Given : 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.

To Find : How many digits does this number N have

Solution:

N is Divisible by 12  prime numbers  and N is odd number hence N can not have 2

so  12 prime numbers will be

3 , 5 , 7 , 11, 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41

x = product of these Numbers

N is a square  and all factors are prime

Hence Each factor  must be in pair

N is a cube Hence each factor must be in a group of 3

so Each factor must be in a group of  6   LCM ( 2 , 3)

N = x⁶  = ( x³)² = (x²)³   Hence a square and a cube  as well

N = ( 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 )⁶

N = (15,21,25,13,17,63,605)⁶

It will have 86 digits

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