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A positive integer is called terrific if it has exactly $10$ positive divisors. What is the smallest terrific positive integer?

 Jul 7, 2024
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If a number is written in the form of \(p_1^a p_2^bp_3^c...\), where all p's are prime numners, it will have (a+1)(b+1)(c+1)... factors. Therefore, a terrific number must be writen in the form of either \(p^9\), or \(p_1^1p_2^4\). The smallest possibilities for each respectively are 2^9 or 512 and 3(2^4) or 48. Therefore, 48 is the smallest terrific number.

 Jul 7, 2024

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