Answer to Question #269312 in Calculus for Unknown356697

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 different prime numbers. How many digits does this number N have?

Answer 1

Let's start from given condition that N is divisible by 12 prime number, smallest and it is odd.

So we consider the prime number 3,5,7,11,13,17,19,23,29,31,37,41 so these are first 12 prime numbers.

Now N is a perfect square as well as a perfect cube .

So it will be power of 6.

(Because if a number is perfect square and cube both then power will be 6 example= 2⁶ is a perfect square of 2³and cube of 2² so number =(2^2×3))).

So N will be power of 6 of all these ​​​​​prime numbers multiplication.

So $N= (3×5×7×11×13×17×19×23×29×31×37×41)^6$

So $N =1.23938376929×10^{85}$

so N have 86 digits.