Question

There is a number less than 100. The factors of this number are 2,3,4. If the sum of the digits is 15 ,find the number

Answer 1

Answer:

Any integer, n , with an odd number of factors must be a perfect square. This is because factors come in pairs (a,b) such that ab=n and a≤n−−√≤b . The only way to get an odd number of factors is if one of the pairs has a=b in which case n is a perfect square.

To have precisely three factors the perfect square must have only one prime factor. If it had two or more prime factors then n=p2q2m2 and would have factors 1 , p , p2 , q , q2 , pq , and more which is too many.

Furthermore it must be the square of a prime, not any higher power. If n=pk with k≥3 then it would have factors 1 , p , p2 , p3 which is too many.

So, we are restricted to squares of prime numbers.

Simply count the primes less than 10 and you have your answer.

Step-by-step explanation:

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