The product of the two-digit numbers above is the three-digit number

SOLUTION

AB
x BA

The product of the two-digit numbers above is the three-digit number ACA, where A, B and C are three different nonzero digits. If A x B <10, what is the two-digit number AB?

(A) 11
(B) 12
(C) 13
(D) 21
(E) 31

First of all, since the digits must be distinct, then we can eliminate option A (11).

Next, simply plug in the options and see which satisfies AB x BA = ACA:

(B) 12 --> 12*21 --> the units digit is not 1. Discard.
(C) 13 --> 13*31 --> the units digit is not 1. Discard.
(D) 21 --> 21*12 = 252 = AB x BA = ACA. Bingo.

Answer: D.
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While plugging answer choices is good way to attack this question, it is good to leverage some facts given in the question stem first, so that we need to try out only a few choices.

1) Units digit of AB*BA is A. That means B*A = A, This is possible when B=1. so we have that A1 * 1A = 1C1
2) Since A, B, and C are DIFFERENT non-zero digits and since B=1, we can say that A is not equal to 1

So we have to check only Choice D and E.

Answer = D = 21

21 (AB) x 12 (BA) = 252 (ACA)

All other options do not satisfy

In Option A, it comes up as A = B & AB x BA = ABA; so ignore it

Option B, C & E do not satisfy

Answer = D

Why is the units digit not 1? Thanks!

I solved it this way:
1) We can eliminate A, C and E
--> A:11: It's said - Three different nonzero digits, so it can't be 11
--> C: Just plug in numbers 13*31= 403 ( It's wrong because 1st and 3rd Digits are the same in the product)
--> E: Same as c 31*13 = 403 ( It's wrong because 1st and 3rd Digits are the same in the product)
So we have just B and D left.

2) If we use the multiplication rules, it is evident that -->
AB
x BA
--------
ACA -------> So, BxA = A so B must be 1; Answer D - 21 is correct

Also am not clear why units digit cannot be 1?

What you need is AB * BA = ACA. You need to find AB. In the product ACA, the unit's digit (A) should be the same as the tens digit of AB (which is also A) i.e. the tens digit of the number you want to find.
You know that 12 * 21 = 252
The units digit of 252 is 2. It should be same as the tens digit of the first number i.e. 12 but the tens digit of 12 is 1.

So the product should instead be written as 21 * 12 = 252. Now, tens digit of AB (i.e. 21) is 2 and units digit of ACA (252) is also 2. They match. So AB = 21 and not 12.

This is what is meant by "units digit is not 1". When you have 12*21, the product's units digit should be the same as the tens digit of 12 so it should be 1. But the product's units digit is not 1; it is 2. Hence, you discard this option.
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[quote="Bunuel"]SOLUTION

AB
x BA

The product of the two-digit numbers above is the three-digit number ACA, where A, B and C are three different nonzero digits. If A x B <10, what is the two-digit number AB?

(A) 11
(B) 12
(C) 13
(D) 21
(E) 31

We are told that all 3 digits are different. So we can reject option (A).

Let us put values and find out the answer.

Option (B) 12
So, 12*21. But here units digit is not A i.e.1. So reject this.
Option (C) 13
So, 13*31. But here units digit is not A i.e.1. So reject this.
Option (D) 21
So, 21*12 = 252 = AB x BA = ACA. This satisfies.

Hence option (D).

--
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AB
*
BA
-----------
ACA

A*B<10 means that B=1 and A can be 2,3,4,5,6,7,8,9. Eliminate A,B,C answers

Test only 21 and 31

21*12=252, it fits

D

Units digit is A and A*B = A => B=1. Also A,B, and C are different digits. Thus, Discard A, B and C choices.

Upon checking 21 we see that it matches the criteria and is, therefore, the answer choice.

Hi,

If this was to be done logically, this is how it would be like

Lets See how two numbers can be multiplied
XY
*YX
--------
\(\hspace{3cm}X^2 \hspace{.3cm} XY\)
\(\hspace{1.3cm} XY \hspace{1cm} Y^2\hspace{.5cm}\) 0
---------------------------------------
\(\hspace{1.3cm} X \hspace{1cm} Z \hspace{.7cm}\) X

Now here we are given certain Conditions , that X and Y are distinct integers and product of XY < 10

The Units Place of product is obtained when we do the following : X*Y +0 = X what can we infer . Using the identity that 1*A = A we can say that Y is definitely 1. And 1*X =X

The Hundreds Place of product is obtained when we do the following : XY + any carry forward from ten's place But . Its told to us that its X . Since we have seen that Y is 1 XY is =X if there is no carry forward from Ten's Place.

The Tens Place of product is obtained when we do the following : \(X^{2}\) +\(Y^{2}\) = Z

But here we Know that Y =1 So \(X^{2}\) +1 gives us the value of Z. Also Since tens Place is a number <10, because any double digit number would lead to carry forward to hundreds place the only integer whose square is less than 10 is 2,3,
But \(2^{2}\)= 4 , \(3^{3}\)=9, If it were 3 then \(3^{2}\) would be 9 and \(Y^{2}\) =1 which would be 10, this would provide carry to hundreds digit. So X is not 3 , hence X is 2.

So we have X as 2 and Y as 1
which is 21

Probus
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Hi Bunuel
We don't have any clear indication about the "C" in the question stem. So, how do someone convinced that this 5 is C?
Thanks__
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Bunuel EMPOWERgmatRichC
I'm talking about only choice C and E

Multiplying 13 with 31 will give the SAME result like multiplying 31 with 13, which is 403 (Apart from the product ACA and non-zero number)
Note: considering AB x BA
Two answer choice CAN'T be the right answer at a time. So, we can cross out choice C and E.
Is it the right way to eliminate choice C and E?
Thanks__
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Hi Asad,

Based on the logic that you have described, what do you think about Answers B and D (keeping in mind what the correct answer is....)?

With this question, TESTing THE ANSWERS would likely be the fastest approach. Keep in mind that the first digit of the answer represents "A" and the second digit represents "B", so the product (re: ACA) has to begin AND end with the same "A" digit. Recognizing the importance of that detail of the question should allow you to make fairly quick work of the answer choices.

GMAT assassins aren't born, they're made,
Rich

why can't A=1 and B be the variant? Still don't understand

Since all digits are different, Option A is out.
Now let us check for 13 and 31. 13x31 = 403, since this is not of the form ACA we can eliminate them too.
This leaves us with 12 and 21.
12 x 21 = 252
AB x BA = ACA (Not satisfied) which leaves us with the answer 21 which is option D.

option A (11) is out of the race since A and B should be same where the rule is flaunted

Next we are employing brute force method to get to the desired answer choice

(B) 12 --->12*21=>252 since the product A and starting a doesn't match we are discarding

(C) 13 --> 13*31 -->403 none of the digits fit in so eleminate

(D) 21 --> 21*12 = 252 = the product A and the intial value of A(2) match

(E) 31--->31*13---> 403 entirely flaunts the given criteria


Therefore IMO D

Correct option D : 21
What a beautiful problem, really loved soving it.

What we have
1. (AB x BA) = ACA
2. A x B <10
- what is the two-digit number AB?

Syntax : AB x BA = ACA and A>=B, not (A=B)

(A) 11 = 11 x 11 = 121 - matches but A=B, condition fail
(B) 12 = 12 x 21 = 252 - A>= B, but A is 2, which must be 1
(C) 13 = 13 x 31 = 403 - fails both condition
(D) 21 = 21 x 12 = 252 - A location 2, and A>=B
(E) 31 = 31 x 13 = 403 - fails boths condition

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