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How many integers between 18 and 3399 are multiples of 17?

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How many integers between 18 and 3399 are multiples of 17?  [#permalink]

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New post 13 Feb 2019, 01:56
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[GMAT math practice question]

How many integers between \(18\) and \(3399\) are multiples of \(17\)?

\(A. 198\)
\(B. 199\)
\(C. 200\)
\(D. 201\)
\(E. 202\)

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Re: How many integers between 18 and 3399 are multiples of 17?  [#permalink]

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New post 13 Feb 2019, 02:01
First integer,a = 34, difference, d = 17.

3400 is divisible by 17, as 17*200
Then last multiple = 3400-17 = 3383.
3383 = 34+(n-1)17
n-1 = (3383-34)/17 = 197. n = 198

A is the answer

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Re: How many integers between 18 and 3399 are multiples of 17?  [#permalink]

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New post 13 Feb 2019, 02:28
Range = 3399-18+1=3381
Number of multiples of 17 in the range = 3381/17=198.88 = 198 multiples

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Re: How many integers between 18 and 3399 are multiples of 17?  [#permalink]

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New post 13 Feb 2019, 07:44
Top Contributor
MathRevolution wrote:
[GMAT math practice question]

How many integers between \(18\) and \(3399\) are multiples of \(17\)?

\(A. 198\)
\(B. 199\)
\(C. 200\)
\(D. 201\)
\(E. 202\)


Another approach:

Let's list the multiples of 17 that are between 18 and 3399 :
34 = 2(17)
51 = 3(17)
68 = 4(17)
85 = 5(17)
.
.
.
3383 = 199(17)

So, the number of multiples of 17 is EQUAL to the number of integers from 2 to 199 inclusive

ASIDE: A nice rule says: the number of integers from x to y inclusive equals y - x + 1
So, the number of integers from 2 to 199 inclusive = 199 - 2 + 1 = 198

Answer: A

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Brent
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Re: How many integers between 18 and 3399 are multiples of 17?  [#permalink]

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New post 13 Feb 2019, 08:40
MathRevolution wrote:
[GMAT math practice question]

How many integers between \(18\) and \(3399\) are multiples of \(17\)?

\(A. 198\)
\(B. 199\)
\(C. 200\)
\(D. 201\)
\(E. 202\)


17*200 = 3400 = multiple of 17.
3399 is one less than a multiple of 17 and thus cannot itself be a multiple of 17.
Since neither endpoint is a multiple of 17, we can simply divide 17 into the difference between the two endpoints:
\(\frac{3399-18}{17} = \frac{3381}{17} = 198\frac{15}{17}\).
Since 17 divides into the difference 198 times, there are 198 multiples of 17 between the two endpoints.


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Re: How many integers between 18 and 3399 are multiples of 17?  [#permalink]

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New post 15 Feb 2019, 01:20
=>

\(18 = 17*1 + 1, 34 = 17*2\) is the least multiple of \(17\) greater than \(18\).
\(3399 = 17*199 + 16, 3383 = 17*199\) is the greatest multiple of \(17\) less than \(3399.\)
Thus, the multiples of \(17\) between \(18\) and \(3399\) are \(34=17*2, … , 3383 = 17*199.\) There are \(198 = (199 – 2) + 1\) of these numbers as they are in one-to-one correspondence with the integers between \(2\) and \(199\), inclusive.

Therefore, the answer is A.
Answer: A
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Re: How many integers between 18 and 3399 are multiples of 17?  [#permalink]

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New post 15 Feb 2019, 18:21
MathRevolution wrote:
[GMAT math practice question]

How many integers between \(18\) and \(3399\) are multiples of \(17\)?

\(A. 198\)
\(B. 199\)
\(C. 200\)
\(D. 201\)
\(E. 202\)


Notice that 17 is a multiple of 17. However, since 17 < 18, we need to go to the next one, 34, as the first multiple of 17 that is within the range of 18 to 3399. Similarly, notice that 3400 is a multiple of 17. However, since 3400 > 3399, we need to go to the previous one, 3383, as the final multiple of 17 that is within the range of 18 to 3399. Therefore, the number of multiples of 17 within the given range is:

(3383 - 34)/17 + 1 = 197 + 1 = 198

Answer: A
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Re: How many integers between 18 and 3399 are multiples of 17?   [#permalink] 15 Feb 2019, 18:21
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