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# If X = 373^p * 423^q, where p and q are positive integers, what is

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Manager
Joined: 13 May 2017
Posts: 103
Location: Finland
Concentration: Accounting, Entrepreneurship
GMAT 1: 530 Q42 V22
GPA: 3.14
WE: Account Management (Entertainment and Sports)
If X = 373^p * 423^q, where p and q are positive integers, what is  [#permalink]

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10 Oct 2018, 02:00
00:00

Difficulty:

55% (hard)

Question Stats:

63% (02:12) correct 37% (02:02) wrong based on 19 sessions

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$$If X = 373^p * 423^q$$, where p and q are positive integers, what is the units digit of X?
(1) $$p^2 + 2pq + q^2 = 36$$
(2) p = 2
Intern
Joined: 01 Sep 2018
Posts: 5
Location: India
Schools: ISB '20
GMAT 1: 710 Q49 V38
GPA: 2.9
Re: If X = 373^p * 423^q, where p and q are positive integers, what is  [#permalink]

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10 Oct 2018, 02:37
1
Divisibility question which deals with the units digit. (There are some that deal with the tens digit as well)
So we will only look at the units digits of the question. Both 373 and 423 end with a 3. Thus list out all the units digit that occur with different powers of 3:
3^1 = 3
3^2 = 9
3^3= 27
3^4= 81
3^5= 243
we stop here because we can see that after this the units digit is repeating (it becomes 3 again, same as in 3^1).

Now on to the statements.
1. says that p^2 + 2pq + q^2 = 36
Thus (p+q)^2 = 36 and thus p+q = +6/-6. It says in the question that p and q are positive integers So p + q cannot be -6

Thus possible values of p and q are
p q
1 5 : Units digit of 3^1 =3 and 3^5 =3. thus Units digit of question stem will be 3 x 3= 9
2 4 : Same as above, units digit will be 9 and 1 respectively and thus 9 x 1 = 9
3 3 : 7 and 7 respectively, and 7 x 7 = 49 and thus again units digit is 9
4 2 : No need to calculate for this as it will be same as 2,4
5 1 : No need to calculate

Thus we can see for given values of p and q the answer is always the same i.e 9. so 1. is sufficient

2. Given that only p = 2, q can take any value. Thus it is clearly insufficient

The answer is thus A

The main trick in the question was to realise that even after multiplying two different numbers we are getting the same answer of 9. Most people after finding that there can be multiple values of p and q would have deemed 1. as insufficient and jumped to the conclusion that you need both 1. and 2. to solve the question and thus get the wrong answer as C. Identifying tricks that the gmat uses is essential for scoring 700+
Re: If X = 373^p * 423^q, where p and q are positive integers, what is   [#permalink] 10 Oct 2018, 02:37
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# If X = 373^p * 423^q, where p and q are positive integers, what is

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