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Director  Joined: 03 Sep 2006
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What is the value of length n<100 meter of wire?  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 68% (01:56) correct 32% (01:46) wrong based on 459 sessions

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What is the value of length n<100 meter of wire?

(1) When divided equal part of 10 meter each, a piece of 5 meter is left.

(2) When divided equal of 6 meter each, a piece of 1 meter is left.
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Joined: 02 Sep 2009
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LM wrote:
What is the value of length n<100 meter of wire?

1) When divided equal part of 10 meter each, a piece of 5 meter is left.

2) When divided equal of 6 meter each, a piece of 1 meter is left.

What is the value of length n<100 meter of wire?

(1) When divided equal part of 10 meter each, a piece of 5 meter is left --> $$n=10q+5$$: 5, 15, 25, ..., 95. Not sufficient.

(2) When divided equal of 6 meter each, a piece of 1 meter is left --> $$n=6p+1$$: 1, 7, 13, 19, 25, ..., 97. Not sufficient.

(1)+(2) General formula of $$n$$ based on $$n=10q+5$$ and $$n=6p+1$$ would be $$n=30m+25$$ --> $$n$$ can be: 25, 55 or 85. Not sufficient.

To elaborate more. How to derive general formula of $$n$$ based on $$n=10q+5$$ and $$n=6p+1$$: divisor will be the least common multiple of above two divisors 6 and 10, hence 30. Remainder will be the first common integer in above two patterns, hence 25. So, to satisfy both conditions, $$n$$ must be of a type $$n=30m+25$$: 25, 55 or 85.

manhattan-remainder-problem-93752.html#p721341
when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html#p722552
when-the-positive-integer-a-is-divided-by-5-and-125591.html#p1028654

Hope it helps.
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Manager  Joined: 12 Nov 2011
Posts: 54
Re: What is the value of length n<100 meter of wire?  [#permalink]

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clear explanation
Director  Joined: 03 Sep 2006
Posts: 600

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Bunuel wrote:
LM wrote:
What is the value of length n<100 meter of wire?

1) When divided equal part of 10 meter each, a piece of 5 meter is left.

2) When divided equal of 6 meter each, a piece of 1 meter is left.

What is the value of length n<100 meter of wire?

(1) When divided equal part of 10 meter each, a piece of 5 meter is left --> $$n=10q+5$$: 5, 15, 25, ..., 95. Not sufficient.

(2) When divided equal of 6 meter each, a piece of 1 meter is left --> $$n=6p+1$$: 1, 7, 13, 19, 25, ..., 97. Not sufficient.

(1)+(2) General formula of $$n$$ based on $$n=10q+5$$ and $$n=6p+1$$ would be $$n=30m+25$$ --> $$n$$ can be: 25, 55 or 85. Not sufficient.

To elaborate more. How to derive general formula of $$n$$ based on $$n=10q+5$$ and $$n=6p+1$$: divisor will be the least common multiple of above two divisors 6 and 10, hence 30. Remainder will be the first common integer in above two patterns, hence 25. So, to satisfy both conditions, $$n$$ must be of a type $$n=30m+25$$: 25, 55 or 85.

manhattan-remainder-problem-93752.html#p721341
when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html#p722552
when-the-positive-integer-a-is-divided-by-5-and-125591.html#p1028654

Hope it helps.

To elaborate more. How to derive general formula of $$n$$ based on $$n=10q+5$$ and $$n=6p+1$$: divisor will be the least common multiple of above two divisors 6 and 10, hence 30. Remainder will be the first common integer in above two patterns, hence 25. So, to satisfy both conditions, $$n$$ must be of a type $$n=30m+25$$: 25, 55 or 85.

Thanks very much. Above would have been very difficult to figure out in the real exam.
Director  Joined: 03 Aug 2012
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Re: What is the value of length n<100 meter of wire?  [#permalink]

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N<100

What is N?

(1).

N =10A +5 ..... N can be 5,15,25,35 and so on

INSUFFICIENT

(2).

N= 6B + 1 .... N can be 1,7,13,19,25 and so on

INSUFFICIENT

Combining (1).& (2).

We get N = 30X + 25

N can be 25,55,85

Hence INSUFFICIENT

(E) it is !!
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Re: What is the value of length n<100 meter of wire?  [#permalink]

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Actually, this problem you can solve at most in 10 seconds:)

The main point here is that all numbers with exact remainder form arithmetic progression with difference=divisor.

For example, all x such that "when x is divided by 5 the remainder is 1" form arithmetic progression with first element 1 and difference 5:
1, 6, 11, 16, 21.....

If 50<x<100 for example, I can definitely say that there are several such x, because the distance between all such numbers is 5.

To solve this problem you need just to check if the divisor=(distance between numbers) large enough to have only 1 number inside interval.

So, I need to find exact number less than 100.
(1) The difference=divisor=10 is quite small for 100. Insufficient.
(2) The difference=divisor=6 is quite small for 100. Insufficient.

(1)+(2) The new difference=least common multiple of 10 and 6=30 is small for 100. Insufficient.

You don't really need here to write formula for x and first several values for each statement.
Intern  Joined: 17 Feb 2015
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Re: What is the value of length n<100 meter of wire?  [#permalink]

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smyarga wrote:
Actually, this problem you can solve at most in 10 seconds:)

The main point here is that all numbers with exact remainder form arithmetic progression with difference=divisor.

For example, all x such that "when x is divided by 5 the remainder is 1" form arithmetic progression with first element 1 and difference 5:
1, 6, 11, 16, 21.....

If 50<x<100 for example, I can definitely say that there are several such x, because the distance between all such numbers is 5.

To solve this problem you need just to check if the divisor=(distance between numbers) large enough to have only 1 number inside interval.

So, I need to find exact number less than 100.
(1) The difference=divisor=10 is quite small for 100. Insufficient.
(2) The difference=divisor=6 is quite small for 100. Insufficient.

(1)+(2) The new difference=least common multiple of 10 and 6=30 is small for 100. Insufficient.

You don't really need here to write formula for x and first several values for each statement.

For (1)+(2), we need to know that the first number is 25. Only then we could say that this is insufficient. If the first number was >70, (1)+(2), could've been sufficient.
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Re: What is the value of length n<100 meter of wire?  [#permalink]

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LM wrote:
What is the value of length n<100 meter of wire?

(1) When divided equal part of 10 meter each, a piece of 5 meter is left.

(2) When divided equal of 6 meter each, a piece of 1 meter is left.

n < 100 ----- (1)

S1- When divided equal part of 10 meter each, a piece of 5 meter is left.

implies n = 10k + 5, where k is some integer

10k + 5 < 100 (from 1)

solving k < 9.5 implies no unique solution for n as it can have multiple values. Therefore insufficient.

S2- When divided equal part of 6 meter each, a piece of 1 meter is left.

solving as above we get k < 16.5 which again means it is insufficient.

Even combining S1 and S2 we don't reach a definite solution. Hence answer is E.
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Re: What is the value of length n<100 meter of wire?  [#permalink]

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LM wrote:
What is the value of length n<100 meter of wire?

(1) When divided equal part of 10 meter each, a piece of 5 meter is left.

(2) When divided equal of 6 meter each, a piece of 1 meter is left.

n=10q+5
n=6p+1
→10q-6p=-4
least values of q and p are 2 and 4 respectively
substituting, least value of n=25
lcm of divisors 10 and 6=30
so 3 possible <100 meter values of n: 25, 55, 85
neither 1 nor 2 sufficient
E Re: What is the value of length n<100 meter of wire?   [#permalink] 24 Jan 2020, 10:17
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