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What is the smallest positive integer with exactly 14 positive divisors?

Nov 2, 2019

#1
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2^6 x 3 = 192 =(1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192) >>Total = 14

Nov 2, 2019
#2
+2833
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Guest, why does doing 26 * 3 give 14 positive divisors?

CalculatorUser  Nov 3, 2019
#3
+108629
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Thanks Guest,

CU's question,

why does doing 2^6 * 3 give 14 positive divisors?

Just count them

1,2,4,8,16,32 64    that is all the factors of 2^6

so that is 7 factor

then

1,3 are the only factors of 3  and you already have 1

so that is another factor, making 8 factors,

Then

2*3=6

4*3=12

8*3=24

16*3=48

32*3=96

64*3=192

So that is another 6

Making 8+6=14 factors altogether.

So 2^6*3 defiitely has 14 factors but I am not necessarily convinced that it is the smallest number with 14 factors.

Nov 3, 2019
edited by Melody  Nov 3, 2019
#4
+1

The number of divisors = The product of the exponents of prime factors + 1 for each prime factor. That is: 2^6 x 3^1 =[6 +1] x [1 + 1] =7 x 2 = 14 divisors. Therefore, 2^6 x 3^1 = 192 is the smallest number with EXACTLY 14 divisors.

Nov 3, 2019
#5
+108629
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ok thanks, I shall try and remember.

since 2*7  and 1*14 are the only integer factors of 14 .

It does follow that 192 must be the smallest one.

Melody  Nov 3, 2019
edited by Melody  Nov 3, 2019