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14 Apr 2020, 04:35
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A
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
65% (01:22) correct
35%
(01:55)
wrong
based on 398
sessions
History
Date
Time
Result
Not Attempted Yet
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
A. 180
B. 216
C. 240
D. 256
E. None of the above
Are You Up For the Challenge: 700 Level Questions
14 Apr 2020, 04:56
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Bunuel
If \(N = a^p*b^q*c^r...\)
where a, b, c... are distinct primes
Number of factors of \(N = (p+1)*(q+1)*(r+1)*...\)
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
-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
Answer: Option A
General Discussion
14 Apr 2020, 12:48
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Bunuel
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.
Answer is A.
Feel free to add kudos if you like the explanation.
