Question

A number greater than 1000 whose prime factorization contains one prime number that does not repeat one prime number that repeats three times and one prime number that repeats twice

Answer 1

Consider the prime factorization of 1000:

$1000=10\cdot 10\cdot10=(2\cdot5)\cdot(2\cdot5)\cdot(2\cdot5)=2^3\cdot5^3$

We see that the prime factorization of 1000 contains 5 repeating three times and 2 repeating three times.

We can keep 5 repeating 3 times, 

We can write a prime whose square is larger than $2^3$, for example: $7^2$

Then we can write any prime which does not repeat, for example 2.


Thus the number we formed is $2 \cdot7^2\cdot5^3=12,250$


Answer: 12,250