Mathematics Asked by AdhirajSinghBrar on December 25, 2020

We know that after the prime factorization of that number when the powers of those prime factors +1 when multiplied with each other would result in 144.

144 = (a+1)(b+1). . .

We also know that there are 10 consecutive numbers among these divisors so the number must be divisible by 2,3,5,7.

I don’t know what to do now , what to do ?

What you say is true, but you should apply it. Factor $144$. You know that there are at least $4$ distinct prime factors. There are not many patterns of exponents of primes to choose from, then not many ways to choose the primes so they include $2,3,5,7$. Make a list of the patterns of exponents, then the smallest numbers that satisfy each pattern. Sort them and list the divisors of each, starting from the smallest. I suspect you will not need to search far. What is the factorization of $LCM(1,2,3ldots 10)?$ What is the minimum you need to multiply it by to get $144$ distinct factors? That gives an upper bound to the answer.

Correct answer by Ross Millikan on December 25, 2020

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