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# If a, b, and c represent distinct digits, what is the value of the two

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Math Expert
Joined: 02 Sep 2009
Posts: 49271
If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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21 Mar 2017, 04:49
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95% (hard)

Question Stats:

47% (01:55) correct 53% (02:31) wrong based on 124 sessions

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If a, b, and c represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

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Posts: 984
Re: If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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21 Mar 2017, 11:49
Bunuel wrote:
If a, b, and b represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

(1) b^2=c

if b =-2 , c= 4
if b= -3 , c=9
a could be 5 or 6
not suff

(2)again same as (1)
insuff

combining same as above
insuff

Ans E
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Re: If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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21 Mar 2017, 23:00
rohit8865 wrote:
Bunuel wrote:
If a, b, and b represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

(1) b^2=c

if b =-2 , c= 4
if b= -3 , c=9
a could be 5 or 6
not suff

(2)again same as (1)
insuff

combining same as above
insuff

Ans E

I believe that you are misinterpreting the question. a, b, and c represent digits, and ab, ba, and aca represent two two-digit numbers and a three-digit number. These are not products. Like, for instance, if a = 8 and b = 3, then ab would be 83, ba would be 38, etc.
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Re: If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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22 Mar 2017, 00:58
1
1.suff: 21 12 only combo acceptable as digits are different
2. insuff

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Re: If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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22 Mar 2017, 02:02
Bunuel wrote:
If a, b, and b represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

St 1: Nothing can be predicted. INSUFFICIENT

St 2: (a,b)/(b,a)can be (1,2)(1,3)(1,4).......(1,9), (2,3),(2,4). INSUFFICIENT

St 1 & St 2: for each possible set we will calculate ab*ba, that will be 252, 403, 574, 765,976....

only 12*21 = 252

hence ab can be 12 or 21. INSUFFICIENT

Option E
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Re: If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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22 Mar 2017, 06:21
1
ByjusGMATapp wrote:
Bunuel wrote:
If a, b, and b represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

St 1: Nothing can be predicted. INSUFFICIENT

St 2: (a,b)/(b,a)can be (1,2)(1,3)(1,4).......(1,9), (2,3),(2,4). INSUFFICIENT

St 1 & St 2: for each possible set we will calculate ab*ba, that will be 252, 403, 574, 765,976....

only 12*21 = 252

hence ab can be 12 or 21. INSUFFICIENT

Option E

ab can not be 12, only 21 because aca is 252
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Joined: 25 Dec 2016
Posts: 18
Re: If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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22 Mar 2017, 07:28
rt9 wrote:
ByjusGMATapp wrote:
Bunuel wrote:
If a, b, and b represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

St 1: Nothing can be predicted. INSUFFICIENT

St 2: (a,b)/(b,a)can be (1,2)(1,3)(1,4).......(1,9), (2,3),(2,4). INSUFFICIENT

St 1 & St 2: for each possible set we will calculate ab*ba, that will be 252, 403, 574, 765,976....

only 12*21 = 252

hence ab can be 12 or 21. INSUFFICIENT

Option E

ab can not be 12, only 21 because aca is 252

11 and 11 is another value
Math Expert
Joined: 02 Sep 2009
Posts: 49271
If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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22 Mar 2017, 08:58
vivek2k70 wrote:
rt9 wrote:
ByjusGMATapp wrote:
St 1: Nothing can be predicted. INSUFFICIENT

St 2: (a,b)/(b,a)can be (1,2)(1,3)(1,4).......(1,9), (2,3),(2,4). INSUFFICIENT

St 1 & St 2: for each possible set we will calculate ab*ba, that will be 252, 403, 574, 765,976....

only 12*21 = 252

hence ab can be 12 or 21. INSUFFICIENT

Option E

ab can not be 12, only 21 because aca is 252

11 and 11 is another value

It is mentioned in the stem that a, b, and c represent distinct digits.
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Re: If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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22 Mar 2017, 10:25
3
I am sure, you wanted to mention: If a, b, and C represent distinct digits
Bunuel wrote:
If a, b, and b represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

We know a,b,c are different digits.

ab*ba = aca

So from the 2 digit multiplication process, first we will multiply a*b. As we get the unit digit number as a itself, means b = 1.

just plugging the numbers 21 & 12 gives 252, means a=2, b=1, c=5.

Also there is no other integer which gives mutiplication X1 * 1X = X_X
A- sufficient.
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Joined: 02 Sep 2009
Posts: 49271
Re: If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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22 Mar 2017, 10:34
RMD007 wrote:
I am sure, you wanted to mention: If a, b, and C represent distinct digits
Bunuel wrote:
If a, b, and b represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

We know a,b,c are different digits.

ab*ba = aca

So from the 2 digit multiplication process, first we will multiply a*b. As we get the unit digit number as a itself, means b = 1.

just plugging the numbers 21 & 12 gives 252, means a=2, b=1, c=5.

Also there is no other integer which gives mutiplication X1 * 1X = X_X
A- sufficient.

Sure. Edited. Thank you for noticing. +1.
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If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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22 Mar 2017, 23:15
ByjusGMATapp wrote:
Bunuel wrote:
If a, b, and b represent distinct digits, what is the value of the two-digit number ab?

(1) ab*ba = aca

(2) a*b < 10

St 1: Nothing can be predicted. INSUFFICIENT

St 2: (a,b)/(b,a)can be (1,2)(1,3)(1,4).......(1,9), (2,3),(2,4). INSUFFICIENT

St 1 & St 2: for each possible set we will calculate ab*ba, that will be 252, 403, 574, 765,976....

only 12*21 = 252

hence ab can be 12 or 21. INSUFFICIENT

Option E

ByjusGMATapp plz elaborate the stat1 ,, according to me stat 1 is suff and 12,21 is the only combo tat fits in.. thanks
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If a, b, and c represent distinct digits, what is the value of the two  [#permalink]

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11 Jan 2018, 12:29
1
1
Statement 1:
$$ab$$ * $$ba$$ = $$aca$$
=> $$(10a+b)(10b+a)$$ = $$100a + 10c + a$$
=> $$100ab + 10a^2 + 10b^2 + ab$$ = $$100a + 10c + a$$
=> $$100ab + (a^2 + b^2)10 + ab$$ = $$100a + 10c + a$$

now comparing units digit on both the sides
$$ab$$= a => $$a= 0$$ OR $$b = 1$$ => but a cannot be 0, if it were, then aca would be 2-digit number => b = 1

now comparing tens digit on both the sides
$$a^2 + b^2 = c$$
$$a^2 = c - b^2$$
$$a^2 = c - 1$$
$$a$$ = $$\sqrt{c - 1}$$
only for c = 1, 2, 5, $$\sqrt{c - 1}$$ yields integer
if c = 1, a = 0 (which is not possible)
if c = 2, a = 1 (but b is already 1) => not possible
so only c = 5, a = 2, possible
so number ab = 21 => sufficient

Statement 2: a * b < 10
lot of combinations ab = {14, 21, 24} not sufficient

(A)
If a, b, and c represent distinct digits, what is the value of the two &nbs [#permalink] 11 Jan 2018, 12:29
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