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Math Expert V
Joined: 02 Sep 2009
Posts: 55801
The product of the units digit, the tens digit, the hundreds digit, an  [#permalink]

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1
5 00:00

Difficulty:   85% (hard)

Question Stats: 46% (02:09) correct 54% (02:00) wrong based on 146 sessions

HideShow timer Statistics The product of the units digit, the tens digit, the hundreds digit, and the thousands digit of the positive integer k is 54. What is the tens digit of k?

(1) k is even

(2) The tens digit of k is 1 larger than the units digit of k

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Math Expert V
Joined: 02 Aug 2009
Posts: 7763
Re: The product of the units digit, the tens digit, the hundreds digit, an  [#permalink]

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Bunuel wrote:
The product of the units digit, the tens digit, the hundreds digit, and the thousands digit of the positive integer k is 54. What is the tens digit of k?

(1) k is even

(2) The tens digit of k is 1 larger than the units digit of k

so we look at the ways we can factor 54 in four single digit factors..
54 = 1*1*6*9..
54 = 1*2*3*9..
54 = 2*3*3*3

lets see the statements--

(1) k is even
so number can be ..
1196
1916
1932
1392
different tens digit
Insuff

(2) The tens digit of k is 1 larger than the units digit of k
the number can be
9321
9132..
again different tens digit
Insuff

combined..
the last digit has to be 2 and the tens digit is ONE more than the units digit..
so tens digit = 2+1 = 3
suff
C
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Intern  Joined: 06 Apr 2016
Posts: 3
Re: The product of the units digit, the tens digit, the hundreds digit, an  [#permalink]

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The product of the units digit, the tens digit, the hundreds digit, and the thousands digit of the positive integer k is 54. What is the tens digit of k?

(1) k is even

(2) The tens digit of k is 1 larger than the units digit of k

->
Step 1 : Extract the information from question stem :
54 can be represented as :
|
|__ 1 * 54 = 1 * 9 * 6 = 1 * 3 * 3 * 6 or 1 * 9 * 3 * 2
|
|__ 2 * 27 = 2 * 9 * 3 = 2 * 3 * 3 * 3
|
|__ 3 * 18 = 3 * 9 * 2 = 3 * 3 * 3 * 2
= 3 * 6 * 3 = 3 * 3 * 2 * 3
Different possibilities are
1336
1932
2333

Prime Factors of 54 are 3 & 2

Step 2 : State the goal :
We need to find the tens digit of K ?

Step 3 : Satisfy the fact condition and try to disprove or to find the results for question stem

1. K is even so units digit must be any of 2, 4, 6, 8
: 54 is not divisible by 4 & 8 so no combination will yield 4 or 8 so exclude
if tens digit is 2 then k can be 1932
if tens digit is 6 then k can be 1336
Two different solution satisfying the fact. Not sufficient

2. Tens digit is 1 larger than the units digit
From different possibilities we can see 1932 or 9321 can yield two different possibilities. Not sufficient

Both 1 & 2 : K is even and tens digit 1 larger than units digit : 1932 is the only combination which satisfies the both the fact and yields the answer for question.

Step 4 : Answer choice C is correct
BCE
Intern  Joined: 24 Aug 2016
Posts: 9
Schools: Rotman '18
Re: The product of the units digit, the tens digit, the hundreds digit, an  [#permalink]

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let the number be - abcd
a*b*c*d = 54
1) d is even i.e d can be 2 4 6 8 but 4 and 8 are not factors of 54 so we are left with 2 and 6 .. still we can not say anything about c.
insufficient
2)c = d+1
d can be 1,2,3,9,6
so if d is 1 c can be 2 but if d is 2 c can be 3
Insufficient
1)+2)
we know d is either 2 or 6 so c can be 7 or 3 but 7 is not a factor of 54
so we definitely know that c is 3
Sufficient
Math Expert V
Joined: 02 Sep 2009
Posts: 55801
Re: The product of the units digit, the tens digit, the hundreds digit, an  [#permalink]

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Bunuel wrote:
The product of the units digit, the tens digit, the hundreds digit, and the thousands digit of the positive integer k is 54. What is the tens digit of k?

(1) k is even

(2) The tens digit of k is 1 larger than the units digit of k

VERITAS PREP OFFICIAL SOLUTION:

You need four digits that multiply to 54, so begin by factoring 54 with a factor tree:

54 = 2 x 3 x 3 x 3

Since 54 is composed of the prime factors 2, 3, 3, and 3, you might be tempted to conclude that the four digits of the integer must themselves be 2, 3, 3, and 3. For example:

3 3 2 3
or
3 3 3 2
However, other options exist. If you multiply 2 by 3 we could have 6 as one of the digits and the remaining digit could be 1:

6 3 3 1
Or, you could multiply 3 by 3 giving 9 as one of the digits:

3 9 1 2
Statement 1: Since k is even, the units digit of k must be even. The units digit of k could be 2 or 6, giving possibilities such as the following:

3 3 3 2
3 3 1 6
1 1 9 6
The tens digit could then be 1, 3, or 9, so Statement 1 is insufficient.

Statement 2: Search for digit pairs that are one unit apart. 2 and 3 are one such pair, as is 1 and 2. Examples include:

3 3 3 2 -> tens digit is 1 larger than units digit
3 9 2 1 -> tens digit is 1 larger than units digit
The tens digit could either be 2 or 3. INSUFFICIENT

Together: Combining the two statements, you should see that 1932 and 3332 are the only possibilities, thus the correct answer is C.
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Re: The product of the units digit, the tens digit, the hundreds digit, an  [#permalink]

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Bunuel wrote:
The product of the units digit, the tens digit, the hundreds digit, and the thousands digit of the positive integer k is 54. What is the tens digit of k?

(1) k is even

(2) The tens digit of k is 1 larger than the units digit of k

54 = 3^3 * 2
{1, 2, 3, 9}, {1, 1, 6, 9}

Statement 1: Number = 3912 or 9132 i.e. different tens digit
NOT SUFFICIENT

Statement 2: Number = 3921 or 9132 i.e. different tens digit
NOT SUFFICIENT

Combining The two statements
Number = xx32
Sufficient

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