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If a number p is prime, and 2p+1 = q, where q is also prime, then the

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If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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New post Updated on: 26 Feb 2015, 10:18
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If a number p is prime, and 2p+1 = q, where q is also prime, then the decimal expansion of 1/q will produce a decimal with q-1 digits. If this method produces a decimal with 166 digits, what is the units digit of the product of p and q?

A. 1
B. 3
C. 5
D. 7
E. 9

Originally posted by sajib2126 on 26 Feb 2015, 10:15.
Last edited by Bunuel on 26 Feb 2015, 10:18, edited 1 time in total.
Renamed the topic and edited the question.
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Re: If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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New post 26 Feb 2015, 11:45
2
sajib2126 wrote:
If a number p is prime, and 2p+1 = q, where q is also prime, then the decimal expansion of 1/q will produce a decimal with q-1 digits. If this method produces a decimal with 166 digits, what is the units digit of the product of p and q?

A. 1
B. 3
C. 5
D. 7
E. 9

Dear sajib2126,
I'm happy to respond. Something is suspect about this question. I don't know whether this question, as it stands, is a Veritas question, or whether the wording has been changed from the original, but the wording is a bit off.

First of all, any prime number (other than 2 or 5) will have a reciprocal with an infinite number of digits. When the question says that the decimal of 1/q has (q-1) digits, they must mean that a sequence of (q-1) digits repeats. In other words, the infinite decimal has a period of (q - 1)

For example, 1/7 has a sequence of six digits that repeat over and over, like mathematical wallpaper.
1/7 = 0.142857 142857 142857 142857 142857 142857 142857 142857 .....

When we take the reciprocal any prime number J, and look at the decimal expansion, the period might be (J - 1) or any factor of (J - 1). For example, 1/13 has a period of 6, and 1/31 has a period of 15. BTW, these are NOT facts you need to have memorized for the GMAT.

This problem appears to be talking about such periods of the decimal expansions. They are telling us that the prime number q has a reciprocal whose decimal expansion has the maximum period, (q - 1). Since this period is 166, it must be that q = 167. (BTW, I verified with Wolfram Alpha that 1/167 indeed does have a decimal expansion with a period of 166.) Then

q = 2p + 1 = 167
2p = 166
p = 83

So the two numbers concerned are p = 83 and q = 167. Now, for the units digits part, see:
http://magoosh.com/gmat/2013/gmat-quant ... questions/

Well, since 3*7 = 21, that's enough to guarantee that any number that ends with a 3 times any number that ends in a 7 will have a units digit of 1.

OA = (A)

Does all this make sense?
Mike :-)
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If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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New post 10 Mar 2015, 14:11
1
Based on the stem ("1/q will produce a decimal with q-1 digits" and "If this method produces a decimal with 166 digits"),
I infered that q-1=166. Then q = 167

Substituting q=167 into 2p+1 = q, we get tat p=83.

167*83= ....1, So ANS A

I don't know if it is clear enough as a question (I don't know math well enough to judge in this case) but, after reading it for a second time and getting over the initial shock, it wasn't difficult..
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Re: If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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New post 16 Mar 2015, 09:29
1
sajib2126 wrote:
If a number p is prime, and 2p+1 = q, where q is also prime, then the decimal expansion of 1/q will produce a decimal with q-1 digits. If this method produces a decimal with 166 digits, what is the units digit of the product of p and q?

A. 1
B. 3
C. 5
D. 7
E. 9



should this question should be categorized as a 700 level ?
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Re: If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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New post 10 Mar 2016, 15:01
why it is not q = 166*a+1 where a is any integer....just like for 13, 1/13 has a repeating period of 6 so 13 = 6*2+1

also, will this be a real GMAT question. i have honest doubts!
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Re: If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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New post 10 Mar 2016, 16:24
1
gmatwithpooja wrote:
why it is not q = 166*a+1 where a is any integer....just like for 13, 1/13 has a repeating period of 6 so 13 = 6*2+1

also, will this be a real GMAT question. i have honest doubts!

Dear gmatwithpooja,
I'm happy to respond. :-)

As a general rule, the patterns that prime numbers follow are in no way amenable to simple algebra formula. In fact, the ultimate pattern of prime numbers is the single hardest unsolved problem in modern mathematics, known as the Riemann Hypothesis. This what folks with Ph.D.'s in pure mathematics try to tackle. This is leagues and leagues beyond what you need to know for the GMAT. The only rough-and-ready fact that you should know is that no simple algebra formula in the universe is going to predict how prime numbers behave.

One of the behaviors of prime numbers concerns how many repeating digits are in the decimal representation of their reciprocals. For example

1/3 = 0.3333333... (a repeating pattern one digit long)
1/7 = 0.142857 142857 142857 142857 142857 ... (a repeating pattern 6 digits long)
1/11 = 0.09090909090909... (a repeating pattern 2 digits long)
1/13 = 0.076923 076923 076923 076923 076923 ... (a repeating pattern 6 digits long)
1/17 = 0.0588235294117647 0588235294117647 ... (a repeating pattern 16 digits long)

The pattern is: for prime number P, the decimal representation of the reciprocal could have (P - 1) digits, which would be the maximal possible value, or it could have a number of repeating digits that is any factor of (P - 1).
3 - 1 = 2, and 1 is a factor of 2
7 - 1 = 6, the maximum value
11 - 1 = 10, and 2 is a factor of 10
13 - 1 = 12, and 6 is a factor of 12
17 - 1 = 16, the maximum value
There is no easy algebraic pattern that relates the number of repeating digits to P. Some prime numbers have the maximum value of repeating digits, and others don't. It's very idiosyncratic. All of this is also well beyond what you need for the GMAT.

Again, the one concrete take-away you need is: nothing about prime numbers can be predicted or understood with a simple algebraic formula.

For more information on decimals, including facts you do need for the GMAT, see this blog article:
GMAT Math: Terminating and Repeating Decimals

Does all this make sense?
Mike :-)
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Re: If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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New post 18 Sep 2016, 09:25
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OE by veritas prep:

Correct Answer: (A)

Work backwards here. If the method described here generates a decimal with 166 digits, then q-1 = 166 and q = 167. If q = 2p+1, then 167=2p+1 and p = 83. Thus pq = (83)(167), so the units digits are 3 * 7 = 21, giving us a number that ends in 1.
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Re: If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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New post 26 Oct 2016, 19:26
q-1=166 --> q=167

2p+1 =167
2p=166
p=83

p x q = 167 x 83 --> gives us a unit digit of 1
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Re: If a number p is prime, and 2p+1 = q, where q is also prime, then the  [#permalink]

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Re: If a number p is prime, and 2p+1 = q, where q is also prime, then the &nbs [#permalink] 07 Jul 2018, 10:48
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