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Home » Mathematics » Middle School Classroom Math (Ages 9-13)
 
A problem of divisors
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small_unicorn
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#1
A problem of divisors
Find the smallest number with 28 divisors.
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PostPosted: Sat Nov 07, 2009 3:12 pm  Back to top 
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#2
Solution
The number of divisors of a number is the product of one more than the exponents in its prime factorization, where the same primes are combined with exponents.

For example, the prime factorization of 24 is 2^3 \cdot 3^1. Thus, it has (3 + 1)(1 + 1) = 4 \cdot 2 = 8 factors.

Now, since 28 = 2 \cdot 2 \cdot 7, there are three cases. The exponents of the primes can be: 6 and 1 and 1, 13 and 1, 6 and 3, or just 27.

To minimize the number, we raise the smallest prime to the biggest exponent, the next smallest to the next biggest, and so on.

So for each case, the minimum is:

2^6 \cdot 3^1 \cdot 5^1 = \mathbf{960}

2^{13} \cdot 3^1 = \mathbf{24576}

2^6 \cdot 3^3 = \mathbf{1728}

2^{27} = \mathbf{134217728}

Clearly, the smallest of these is \boxed{960}.
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Last edited by PowerOfPi on Sun Nov 08, 2009 6:24 am; edited 1 time in total 
PostPosted: Sat Nov 07, 2009 3:33 pm  Back to top 
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prezcoin
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#3
I don't get the
2^6 \cdot 3^1 \cdot 5^1 = \mathbf{960} 2^13 \cdot 3^1 = \mathbf{24576} 2^6 \cdot 3^3 = \mathbf{1728} 2^27 = \mathbf{134217....

Were you trying to say:
2^6 \cdot 3^1 \cdot 5^1 = \mathbf{960}

2^{13} \cdot 3^1 = \mathbf{24576}

2^6 \cdot 3^3 = \mathbf{1728}

Just wondering.

2^{27} = \mathbf{134217728}

Clearly, the smallest of these is \boxed{960}.
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PostPosted: Sat Nov 07, 2009 5:16 pm  Back to top 
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pinkmuskrat
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#4
@PowerofPi: make sure you type 2^{13}, not 2^13 when dealing with multi digit exponents in \LaTeX Smile

@prezcoin: next time just use the "quote" button for that type of reply
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PostPosted: Sat Nov 07, 2009 7:04 pm  Back to top 
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PowerOfPi
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#5
Sorry, fixed.
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PostPosted: Sun Nov 08, 2009 6:24 am  Back to top 
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