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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

Problem Solving Question: 174 Category:Arithmetic Operations on rational numbers Page: 85 Difficulty: 600

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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.

Re: The product of the two-digit numbers above is the three-digi [#permalink]

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25 Mar 2014, 13:13

Bunuel wrote:

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.

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

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.
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Re: The product of the two-digit numbers above is the three-digi [#permalink]

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23 May 2014, 08:20

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
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Re: The product of the two-digit numbers above is the three-digi [#permalink]

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30 Mar 2015, 21:17

NAL9 wrote:

Bunuel wrote:

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.

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.

Why is the units digit not 1? Thanks!

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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Re: The product of the two-digit numbers above is the three-digi [#permalink]

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31 Mar 2015, 11:00

[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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Re: The product of the two-digit numbers above is the three-digi [#permalink]

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26 Nov 2017, 16:56

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