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# Is the tens digit of a three-digit positive integer p

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Is the tens digit of a three-digit positive integer p [#permalink]

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17 Jun 2013, 12:28
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Is the tens digit of a three-digit positive integer p divisible by 3?

(1) p-7 is a multiple of 3
(2) p-13 is a multiple of 3
[Reveal] Spoiler: OA

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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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17 Jun 2013, 12:57
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Is the tens digit of a three-digit positive integer p divisible by 3?

(1) p-7 is a multiple of 3 --> $$p=3x+7=3(x+2)+1$$. p is 1 more than a multiple of 3. Not sufficient.

(2) p-13 is a multiple of 3 --> $$p=3y+13=3(y+4)+1$$. p is 1 more than a multiple of 3. Not sufficient.

(1)+(2) Nothing new. For example, consider p=103 for an YES answer or p=112 for a NO answer. Not sufficient.

Hope it's clear.
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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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17 Jun 2013, 13:37
Hey,

The natural initial instinct to this question, I suppose, would be to say that they are insufficient, which as you showed at the bottom, is easy to prove by plugging in some numbers.

I was wondering if there is a more strictly mathematical way of proving this without examples? I.E, solving some system of equations where each digit in the number is a variable?

I'm not sure exactly what I'm picturing, but I just want to see a more concrete method than trying a few examples and seeing that they are insufficient.

Thanks!
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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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18 Jun 2013, 20:18
mattce wrote:
Hey,

The natural initial instinct to this question, I suppose, would be to say that they are insufficient, which as you showed at the bottom, is easy to prove by plugging in some numbers.

I was wondering if there is a more strictly mathematical way of proving this without examples? I.E, solving some system of equations where each digit in the number is a variable?

I'm not sure exactly what I'm picturing, but I just want to see a more concrete method than trying a few examples and seeing that they are insufficient.

Thanks!

Hi Matt,

Both equations can be combined as: p=3*t+100 (3 - LCM of 3 and 3, and 16 first common term of individual equations).
Now the terms are - 16,19....100,103....121,124....130...
Among 3 digit numbers :0 and 3 are divisible by 3, but 2 is not . Hence E.

Last edited by cumulonimbus on 20 Jun 2013, 05:54, edited 1 time in total.

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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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18 Jun 2013, 20:36
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mattce wrote:
Is the tens digit of a three-digit positive integer p divisible by 3?

(1) p-7 is a multiple of 3
(2) p-13 is a multiple of 3

p is a three digit positive integer. We need to figure out whether tens digit is one of 0/3/6/9.

(1) p-7 is a multiple of 3
This is harder to imagine. It's easier to say that p - 1 is a multiple of 3 (since if p - 7 is a multiple of 3, p - 4 and p - 1 must be multiples of 3 too). So p is one more than
a multiple of 3. The tens digit of one more than a multiple of 3 may or may not be 0/3/6/9.
For example:
p - 1 could be 123 or 132 (multiples of 3) etc
p will be 124 or 133 - in one case tens digit is divisible by 3 and in another it is not.

(2) p-13 is a multiple of 3
Notice that this gives the same info as statement 1. We already know that p - 7 is a multiple of 3, so is p- 10 and p - 13. So again, we can write it as p -1 must be a multiple of 3 and that alone is not sufficient.

Both statements give the same info and are not sufficient. So together, they are not sufficient too.

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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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18 Jun 2013, 21:13
cumulonimbus wrote:
Both equations can be combined as: p=39*t+120 (39 - LCM of 13 and 3, and 120 first common term of individual equations)

Could you please elaborate on what you're doing here?
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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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20 Jun 2013, 06:03
mattce wrote:
cumulonimbus wrote:
Both equations can be combined as: p=39*t+120 (39 - LCM of 13 and 3, and 120 first common term of individual equations)

Could you please elaborate on what you're doing here?

Hi Matt,

This is a method given by Bunnel. Basically to combine 2 equations. I have come across a good number of questions in which this method can be used.

Here, P=3x+7 as well as P=3y+13, where x an y are integers.
These can be combined as:
P=LCM(of coefficients of integers)*Integer + first common term of these two equations.

P=3x+7 => 7,10,13,16,19..
P=3y+13 =>13,16,19,22..
so these two equations can be combined as:

P=3q+100 => 19,22, 25, ...100,103...112.....130,133

Tens digit is sometimes divisible by 3 (133) and some times not(112). => hence E.

My post above had some mistakes. I have edited my post.

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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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20 Jun 2013, 08:19
cumulonimbus wrote:
Here, P=3x+7 as well as P=3y+13, where x an y are integers.
These can be combined as:
P=LCM(of coefficients of integers)*Integer + first common term of these two equations.

P=3x+7 => 7,10,13,16,19..
P=3y+13 =>13,16,19,22..
so these two equations can be combined as:

P=3q+100 => 19,22, 25, ...100,103...112.....130,133

I can't see how your final equation P = 3q + 100 follows the structure "LCM*Integer + first common term"

Do you mean first positive common term? In this case, wouldn't the value be 13, with x = 2 and y = 0? Where do you get the 100?

Thanks for any clarification.
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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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20 Jun 2013, 09:39
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mattce wrote:
cumulonimbus wrote:
Here, P=3x+7 as well as P=3y+13, where x an y are integers.
These can be combined as:
P=LCM(of coefficients of integers)*Integer + first common term of these two equations.

P=3x+7 => 7,10,13,16,19..
P=3y+13 =>13,16,19,22..
so these two equations can be combined as:

P=3q+100 => 19,22, 25, ...100,103...112.....130,133

I can't see how your final equation P = 3q + 100 follows the structure "LCM*Integer + first common term"

Do you mean first positive common term? In this case, wouldn't the value be 13, with x = 2 and y = 0? Where do you get the 100?

Thanks for any clarification.

No need to use some magic formulas for this question. The easiest methods are given here: is-the-tens-digit-of-a-three-digit-positive-integer-p-154514.html#p1236901 and here: is-the-tens-digit-of-a-three-digit-positive-integer-p-154514.html#p1237351

Method cumulonimbus is talking about is not good for this particular question.

If interested it's described here: positive-integer-n-leaves-a-remainder-of-4-after-division-by-93752.html and here: when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html

Questions were this method is aplicable:
positive-integer-n-leaves-a-remainder-of-4-after-division-by-93752.html
if-n-is-a-positive-integer-greater-than-16-is-n-a-prime-129829.html
when-positive-integer-x-is-divided-by-5-the-remainder-is-128470.html
when-n-is-divided-by-5-the-remainder-is-2-when-n-is-divided-82624.html
when-positive-integer-n-is-divided-by-5-the-remainder-is-90442.html
when-the-positive-integer-a-is-divided-by-5-and-125591.html
what-is-the-value-of-length-n-100-meter-of-wire-126500.html
a-group-of-n-students-can-be-divided-into-equal-groups-of-126384.html
when-the-positive-integer-a-is-divided-by-5-and-7-the-104480.html
positive-integer-n-leaves-a-remainder-of-4-after-division-by-93752.html
when-positive-integer-n-is-divided-by-3-the-remainder-is-86155.html
what-is-the-remainder-when-n-is-divided-by-131442.html

Hope it helps.
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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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Re: Is the tens digit of a three-digit positive integer p [#permalink]

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