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# What is the smallest positive integer that has exactly 18 positive fac

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Math Expert
Joined: 02 Sep 2009
Posts: 64891
What is the smallest positive integer that has exactly 18 positive fac  [#permalink]

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14 Apr 2020, 03:35
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45% (medium)

Question Stats:

53% (01:17) correct 47% (01:53) wrong based on 78 sessions

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What is the smallest positive integer that has exactly 18 positive factors?

A. 180
B. 216
C. 240
D. 256
E. None of the above

Are You Up For the Challenge: 700 Level Questions

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What is the smallest positive integer that has exactly 18 positive fac  [#permalink]

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14 Apr 2020, 03:56
Bunuel wrote:
What is the smallest positive integer that has exactly 18 positive factors?

A. 180
B. 216
C. 240
D. 256
E. None of the above

Are You Up For the Challenge: 700 Level Questions

If $$N = a^p*b^q*c^r...$$
where a, b, c... are distinct primes
Number of factors of $$N = (p+1)*(q+1)*(r+1)*...$$

STRATEGY:
-For smallest number the Prime number i.e. a, b, c etc must be as small as possible and
- Exponents of Primes should be broken into parts which makes the number smallest
- Bigger prime number must have smaller power to keep number as small as possible

18 = 9*2 = (8+1)*(1+1)

i.e. Number may be $$2^8*3^1 = 768$$

OR

18 = 6*3 = (5+1)*(2+1)

i.e. Number may be $$2^5*3^2 = 288$$

OR

18 = 2*3*3 = (1+1)*(2+1)*(2+1)

i.e. Number may be $$2^2*3^2*5 = 180$$

i.e. smallest such number must be 180

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Re: What is the smallest positive integer that has exactly 18 positive fac  [#permalink]

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14 Apr 2020, 11:48
Bunuel wrote:
What is the smallest positive integer that has exactly 18 positive factors?

A. 180
B. 216
C. 240
D. 256
E. None of the above

Are You Up For the Challenge: 700 Level Questions

I would suggest you go with the option.
1) 180= 2^2 * 3^2 * 5
formula to find total number o factors:
If x^a *y^b, total number of factors are (a+1)*(b+1)

In the ques, total number of factors are (2+1)(2+1)(1+1) = 18.

No need to check other options. Though if you want you may.

Feel free to add kudos if you like the explanation.
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Re: What is the smallest positive integer that has exactly 18 positive fac  [#permalink]

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14 Apr 2020, 19:49
Prime factorization of 18 =2×3×3
Therefore smallest number= 2^2×3^2×5=180

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Re: What is the smallest positive integer that has exactly 18 positive fac  [#permalink]

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17 Apr 2020, 10:30
1
1
Bunuel wrote:
What is the smallest positive integer that has exactly 18 positive factors?

A. 180
B. 216
C. 240
D. 256
E. None of the above

Are You Up For the Challenge: 700 Level Questions

Let’s work the procedure backwards for finding the number of positive factors of a particular integer.

Let n be the smallest positive integer that has exactly 18 positive factors. Since 18 = 2 x 3 x 3 = (1 + 1)(2 + 1)(2 + 1), we want n = a^1 x b^2 x c^2, where a, b, and c are distinct prime numbers. Since we want n as small as possible, we want a, b, and c to be the three smallest prime numbers, i.e., 2, 3, and 5. However, to make n the smallest, we want the largest prime number to have the smallest exponent and the smallest prime number to have the largest exponent. Therefore, n = 5^1 x 3^2 x 2^2 = 5 x 9 x 4 = 180.

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Re: What is the smallest positive integer that has exactly 18 positive fac   [#permalink] 17 Apr 2020, 10:30