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# If the product of the digits of the two-digit positive integer n is 12

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Math Expert
Joined: 02 Sep 2009
Posts: 59590
If the product of the digits of the two-digit positive integer n is 12  [#permalink]

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26 Sep 2018, 05:36
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Difficulty:

75% (hard)

Question Stats:

48% (01:59) correct 52% (01:50) wrong based on 63 sessions

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If the product of the digits of the two-digit positive integer n is 12, what is the value of n?

(1) n can be expressed as the sum of two perfect squares in exactly one way.
(2) n is smaller than 40.

_________________
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If the product of the digits of the two-digit positive integer n is 12  [#permalink]

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26 Sep 2018, 06:46
lets start by listing the factors of 12
1 x 12 = 12
2 x 6 = 12
3 x 4 = 12

since it's a 2-digit integer the 1-12 combination is out. (twelve cannot be 'a digit')
next list the numbers that can be formed with the remaining factors
26
34
43
62
now lets look at 1)
sum of perfect squares in 1 way

26
--> 1 & 5 => 1 & 25 = 26
2 results in 4 and 22 which is not a perfect square
3 results in 9 and 17 which is prime
4 results in 16 and 10, which is not a perfect square
5 puts us back in the first situations

SO: 26 matches, however, we are looking for a single number so none of the others should be possible to be represented as a sum of perfect squares, or should be possible to be represented as different summs of perfect squares
-->looking at 34
1 gives 1 and 33 --> not a perf square
2 gives 4 and 30 --> not a perf square
3 gives 9 and 24 --> not a perf square
4 gives 16 and 18 --> not a perf square
5 gives 25 and 9 --> 2 perfect squares
6 gives 36 which is larger than 34

we found 2 possible n values
1 is not sufficient

2)n is smaller than 40
straight out we can have 26 and 34 --> multiple values = not sufficient

1+2) since the 2 values smaller than 40 are also the values that we found complied to 1), this means we still have 2 possible values

Not sufficient --> E

kudos if it helped
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Joined: 21 Nov 2016
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Re: If the product of the digits of the two-digit positive integer n is 12  [#permalink]

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15 Oct 2018, 06:13
We are given that the product of the digits of a two digit positive integer is 12.
That is say x and y are the digits of two digit positive integer (n = xy), then x*y = 12.
now let us check the no. of factors of 12 = 3*2^2 => 2*3 = 6 factors.
The 6 factors are 1, 2, 3, 4, 6, 12.
So 12 could be obtained by the multiplication of the two single digits in the following ways:
1. 2*6
2. 6*2
3. 3*4
4. 4*3
so, we have 4 ways.

Keeping the above information in mind, let us proceed to the statements.
Statement 1 says that n can be expressed as the sum of two perfect squares in exactly one way.
That is, 2^2 + 3^2 = 13, 3^2+4^2 = 25, so on... this does not conclusively give an answer. Insufficient.
Statement 2 says that the no. n is less than 40. So, n<40, from the stimulus data we get 26, 34 as the possible values, still we do not get a unique value. Insufficient.
Combining both statements give us a lot many values starting from 13 until 40, so, since the combination of the statements is also inconclusive.
Re: If the product of the digits of the two-digit positive integer n is 12   [#permalink] 15 Oct 2018, 06:13
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