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

Question

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

Bunuel · Accepted Answer

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.

Narenn · Answer

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.

TOB2020 · Answer

I am not clear why units digit cannot be 1?

KarishmaB · Answer

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.

Probus · Answer

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

Freddy12 · Answer

Hi  Bunuel 
Can you please share links to similar type of Questions?

Bunuel · Answer

Arithmetic 
 
    Theory    
 
 Mathematical Grouping Symbols 
 GMAT Question Patterns: Arithmetic 
 The Many Meanings of Minus 
 How to Solve Simple Math Equations on the GMAT 
 
    Questions    
 
 GMAT Practice Questions with Fractions and Decimals 
 Addition/Subtraction/Multiplication Tables 
 ALPHAMETIC QUESTIONS ON THE GMAT 
 Symbols Representing Arithmetic Operations Problems 
 Conversion Problems 
 DS Arithmetic Questions 
 PS Arithmetic Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

bumpbot · Answer

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

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