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# If P and Q are positive integers, is (P + 1)(Q + 1) = 12 ?

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Math Expert
Joined: 02 Sep 2009
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If P and Q are positive integers, is (P + 1)(Q + 1) = 12 ?  [#permalink]

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15 Nov 2019, 02:16
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55% (hard)

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70% (01:55) correct 30% (02:37) wrong based on 46 sessions

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If P and Q are positive integers, is $$(P + 1)(Q + 1) = 12$$ ?

(1) $$(P – 1) (Q – 1) = 2$$

(2) $$P^2 + Q^2 = 13$$

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Re: If P and Q are positive integers, is (P + 1)(Q + 1) = 12 ?  [#permalink]

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15 Nov 2019, 06:34
1
If P and Q are positive integers, is $$(P + 1)(Q + 1) = 12$$ ?
$$(P + 1)(Q + 1) = 12=1*12=(0+1)(11+1)$$.. But P and Q are positive...Discard
$$(P + 1)(Q + 1) = 12=2*6=(1+1)(5+1)$$.. P and Q are 1 and 5 in any order...
$$(P + 1)(Q + 1) = 12=3*4=(2+1)(3+1)$$.. P and Q are 2 and 3 in any order...

(1) $$(P – 1) (Q – 1) = 2$$
$$(P - 1)(Q - 1) = 2=2*1=(3-1)(2-1)$$.. So, P and Q are 2 and 3 in any order.....ans is YES
As 2 is prime, no other way that product of two positive integer is 2.
Suff

(2) $$P^2 + Q^2 = 13$$
P an q have to be less than 4 as $$4^2=16$$..
P as 1 gives Q as non-integer
P as 2 gives Q as 3 as $$2^2+3^2=13$$
P as 3 gives Q as 2 as $$3^2+2^2=13$$
Again P and Q are 2 and 3 in any order
Suff

D
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Re: If P and Q are positive integers, is (P + 1)(Q + 1) = 12 ?  [#permalink]

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16 Nov 2019, 00:55
Ooption D: I would suggest as we know they are positive int values . only one possible value fits into them.
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Joined: 18 Jul 2019
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Re: If P and Q are positive integers, is (P + 1)(Q + 1) = 12 ?  [#permalink]

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19 Nov 2019, 03:42
Statement 1 ) (P-1) (Q-1) = 2 ==> EITHER P-1 & Q-1 is a one and the other a 2 with same positive or negative sign.
Therefore, P = +/- 2 if P-1 = (+/-)1 or P= (+/-) 3 if P-1 = +/-2. Similarly Q will have the same values . In any case (P+1)(Q+1) = 12. Hence statement 1 is sufficient.

Statement 2 ) P^2 + Q^2 = 13 . P & Q can hold values of (+/-) 2 and (+/ -) 3. Again (P+1)(Q+1) = 12.

Both statement s are sufficient individually.
Re: If P and Q are positive integers, is (P + 1)(Q + 1) = 12 ?   [#permalink] 19 Nov 2019, 03:42
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