What is the smallest positive number with exactly ten positive integer divisors?
And what is the next one after that?
This brainteaser was written by Derrick Niederman.
Solution: 48, 80.
The smallest number with 10 divisors is 48 = 24 × 3.
In general, the number pa × qb has (a+1)(b+1) divisors if p and q are prime numbers. Because 2 and 3 are the smallest prime numbers, a number of the form 2n × 3 is the smallest number with 2(n + 1)divisors, as follows:
n
2n × 3
Number of divisors of 2n × 3
0
3
2
1
6
4
12
24
8
48
10
From this pattern, it isn’t hard to conclude that the next number with precisely 10 divisors can be obtained by replacing 3 with the next prime number, 5, to get 24 × 5 = 80.
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