SOLUTION: What is the smallest positive integer that has exactly 7 positive divisors?

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Question 987822: What is the smallest positive integer that has exactly 7 positive divisors?
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Let's make a table. The first column represents the number X. For example in row 8, we have X = 8. We're finding the divisors of the number X = 8. The same applies for the other rows as well.

The second column represents the number of divisors (eg: the number 7 has 2 divisors, so a 2 will go in the second column in row 7).

The third column represents the actual list of divisors so you can count yourself to check the value in the second column.

We'll stop once we have the second column have a "7" in it


Xnumber of divisors for number Xlist of divisors for number X
111
221,2
321,3
431,2,4
521,5
641,2,3,6
721,7
841,2,4,8
931,3,9
1041,2,5,10
1121,11
1261,2,3,4,6,12
1321,13
1441,2,7,14
1541,3,5,15
1651,2,4,8,16
1721,17
1861,2,3,6,9,18
1921,19
2061,2,4,5,10,20
2141,3,7,21
2241,2,11,22
2321,23
2481,2,3,4,6,8,12,24
2531,5,25
2641,2,13,26
2741,3,9,27
2861,2,4,7,14,28
2921,29
3081,2,3,5,6,10,15,30
3121,31
3261,2,4,8,16,32
3341,3,11,33
3441,2,17,34
3541,5,7,35
3691,2,3,4,6,9,12,18,36
3721,37
3841,2,19,38
3941,3,13,39
4081,2,4,5,8,10,20,40
4121,41
4281,2,3,6,7,14,21,42
4321,43
4461,2,4,11,22,44
4561,3,5,9,15,45
4641,2,23,46
4721,47
48101,2,3,4,6,8,12,16,24,48
4931,7,49
5061,2,5,10,25,50
5141,3,17,51
5261,2,4,13,26,52
5321,53
5481,2,3,6,9,18,27,54
5541,5,11,55
5681,2,4,7,8,14,28,56
5741,3,19,57
5841,2,29,58
5921,59
60121,2,3,4,5,6,10,12,15,20,30,60
6121,61
6241,2,31,62
6361,3,7,9,21,63
6471,2,4,8,16,32,64



Things to notice
1) The prime numbers only have 2 divisors. This is always true because that's what a prime number is: a number divisible by itself and 1 (and no other number).
2) The perfect squares have an odd number of divisors while any other number has an even number of divisors. So because we want 7 divisors, this means that we could restrict ourselves to just looking at the perfect squares. It makes a much nicer table that is smaller

Xnumber of divisors for number Xlist of divisors for number X
111
431,2,4
931,3,9
1651,2,4,8,16
2531,5,25
3691,2,3,4,6,9,12,18,36
4931,7,49
6471,2,4,8,16,32,64



From either table, we see that the number 64 is the smallest positive integer that has exactly 7 positive divisors (or factors). Those divisors are 1,2,4,8,16,32,64

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If you need more help, or if you have any questions about the problem, feel free to email me at jim_thompson5910@hotmail.com
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