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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # What is the smallest positive integer which when divided by 9, 18, 24

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Math Expert V
Joined: 02 Sep 2009
Posts: 62352
What is the smallest positive integer which when divided by 9, 18, 24  [#permalink]

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Difficulty:   15% (low)

Question Stats: 88% (01:30) correct 12% (02:00) wrong based on 57 sessions

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What is the smallest positive integer which when divided by 9, 18, 24 leaves a remainder of 5, 14 and 20 respectively?

A. 34
B. 52
C. 62
D. 68
E. 72

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Posts: 4551
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Re: What is the smallest positive integer which when divided by 9, 18, 24  [#permalink]

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Top Contributor
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Bunuel wrote:
What is the smallest positive integer which when divided by 9, 18, 24 leaves a remainder of 5, 14 and 20 respectively?

A. 34
B. 52
C. 62
D. 68
E. 72

Rather than try to actually calculate the value needed, it's probably much faster just to test the answer choices from smallest to largest.

A. 34
34 divided by 24 leaves remainder 10 (not 20)
ELIMINATE

B. 52
52 divided by 24 leaves remainder 4 (not 20)
ELIMINATE

C. 62
62 divided by 24 leaves remainder 14 (not 20)
ELIMINATE

D. 68
68 divided by 9 leaves remainder 5 (good)
68 divided by 18 leaves remainder 14 (good)
68 divided by 24 leaves remainder 20 (good)

Cheers,
Brent

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Re: What is the smallest positive integer which when divided by 9, 18, 24  [#permalink]

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Just check for first one - division by 9 results in remainder 5.

from all options, only Option D satisfies. We can skip checking for others.

Option D
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Re: What is the smallest positive integer which when divided by 9, 18, 24  [#permalink]

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in such problems look for difference. all are same. 9-5=4, 18-14=4, 24-20=4.

direct formula = LCM(9,18,24) - 4. = 72 -4 = 68 (D)
Intern  Joined: 26 Aug 2019
Posts: 4
Re: What is the smallest positive integer which when divided by 9, 18, 24  [#permalink]

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Unless you are an expert on LCM / GCD theory, it is better to try out the different answer choices. In this case, it is simple if you look for multiples of answer choice minus the remainder, that is:

34 - 5 = 29; 29 is not a multiple of 9, incorrect answer.
68 - 5 = 63; 63 is multiple of 9; correct answer, if it satisfies the first option, it must satisfy the rest.
Intern  B
Joined: 24 Jul 2019
Posts: 11
Re: What is the smallest positive integer which when divided by 9, 18, 24  [#permalink]

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It is best to use options for your ease.

Try putting values one by one.
Eventually you will lead to option D).

Ans : D)
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Joined: 12 Apr 2019
Posts: 459
Re: What is the smallest positive integer which when divided by 9, 18, 24  [#permalink]

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Let the integer be N. The divisors with which N is being divided are 9, 18 and 24; the respective remainders are 5,14 and 20 respectively.

Observe that there is a constant difference between each divisor and the respective remainder. When this happens, you can write the number N as,
N = LCM (divisors)*k – constant difference, where k is a positive integer
.

The LCM of 9, 18 and 24 is 72. Therefore, the integer N can be written as,
N = 72k – 4 where k is a positive integer and 4 is the constant difference between each divisor and its corresponding remainder.
Since we need the smallest value of N, we take k=1 which gives us N = 68.

The correct answer option is D.

Remember that this is a standard model of questions on LCM. Look out for a common difference between the divisors and the remainders and if it exists, you can use this model to obtain the answer. Also note that since there is a subtraction, the multiplier ‘k’ has to be a positive integer if the question asks you to find the smallest positive integer that satisfies the conditions.

Hope that helps!
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Re: What is the smallest positive integer which when divided by 9, 18, 24  [#permalink]

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Bunuel wrote:
What is the smallest positive integer which when divided by 9, 18, 24 leaves a remainder of 5, 14 and 20 respectively?

A. 34
B. 52
C. 62
D. 68
E. 72

Solution:

We can start with the largest divisor, 24. The positive integers that leaves a remainder of 20 when divided by 24 are:

20, 44, 68, and so on.

Since 20 and 44 are not in the choices, we see that the correct answer must be 68.

(Note: When 68 is divided by 9, 18, and 24, the remainders are 5, 14, and 20. respectively.)

Alternate Solution:

Notice that if n is an integer that produces remainders of 5, 14 and 20 when divided by 9, 18 and 24 respectively; then n + 4 is divisible by 9, 18 and 24. The smallest value of n + 4 is equal to the smallest number divisible by 9, 18 and 24; i.e. the LCM of 9, 18 and 24; i.e. 72. Since the smallest value of n + 4 is 72, the smallest value of n is 72 - 4 = 68.

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