If a and b are positive integers, What is the value of tens digit of a

Question

If a and b are positive integers, What is the value of tens digit of ab * ba ?
1. a is divisible by 5.
2. b is divisible by 2.

AyushAgrawal1005 · Answer

could you please provide the solution?

jpan · Answer

The question can be reduced to finding ten's digit of (ab)^2.
1) a=5k where k is some positive integer. 
This alone will not be sufficient to tell us ten's digit.
2) b=2x where x is some posiitive integer. 
This alone as well, is not sufficient to tell us about the ten's digit.

Taking 1 and 2 together, we know that product is (10kx)^2. 
Since 100*any integer will give ten's digit as 0, we can mark the answer as C.

st1gg3r · Answer

Can someone help me. Below is my working but it seems wrong.

From Statement I we get a=5. It cant be zero otherwise it wont be a 2 digit number.

From Statement II we get b as 2, 4, 6 or 8.

52x25=1300
54x45=2430
56x65=3640
58x85=3930

So even combining wont give us an answer.

CrackverbalGMAT · Answer

The question says that a and b are positive integers. This means that both ab and ba are two-digit numbers with non-zero digits. 
We are to find the tens digit of the product of ab and ba.

From statement I alone, a is divisible by 5. This means that a=5. 
If a=5 and b=1, ab=51 and ba = 15. The product of ab and ba ends with 65. The tens digit is 6.
If a=5 and b=2, ab=52 and ba=25. The product of ab and ba ends with 00. The tens digit is 0.
Statement I alone is insufficient to find a unique value for the tens digit. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, b is divisible by 2. This means that b = 2 or 4 or 6 or 8.
This is insufficient to find the tens digit of the product of ab and ba since we do not know the value of a.
Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we have the following:
From statement I, a=5 and from statement II, b = 2 or 4 or 6 or 8.

Therefore, possible values for ab = 52 or 54 or 56 or 58 AND
Possible values for ba = 25 or 45 or 65 or 85.

There can be more than one value for the tens digit of the product. The combination of statement is insufficient. Answer option C can be eliminated.

The correct answer option is E .

Kritisood , I think you may have to change the OA to E. Please confirm.

Hope that helps!

Kritisood · Answer

Hi, I had marked E as well but the OA was C hence had posted to see where I was going wrong. I guess I was correct in thinking, thanks!

CrackverbalGMAT · Answer

Hello jpan, 
If we assume ab = 37, ba = 73. In this case, it’s wrong to assume that ab*ba =  (ab)^2 . 
Clearly, 37 * 73 is not equal to  (37)^2 . 
In general, it’s not advisable to assume that ab * ba =  (ab)^2 .

Additionally, a and b represent the digits of a number and hence have to be single digits. So, k cannot be any positive integer, it HAS to be 1. Similarly, x cannot be any positive integer, it HAS to be 1 or 2 or 3 or 4.

This question asks us to find the tens digit of the product of ab and ba and hence the combination of statements is not sufficient.

Hope that helps!

jpan · Answer

Hi Aravind, I would agree with you but I the question seems ambiguous to me. Nowhere is it given that a and b are 1 digit numbers and I did not automatically assume that ab meant a number formed by appending b to a. I assumed ab was product of a and b where they can be any integers and thus reduced the question to (ab)^2. I would never assume number * reverse of the number = number square !

Thanks for your solution though

Victz · Answer

C is for this question instead: a^b * b^a

Commendlionel · Answer

The question did not mention anything about  a  and  b  representing digits to a 2-digit number (even though these letters are conventionally used to illustrate that), and I think in the GMAT the question would explicitly state so. Hence, I do not think it is appropriate to assume.

I have treated  a  and  b  as independent integers and can potentially be 1, 2 or even 3-digit integers. We do not know this fact.
However, we can start off by expressing  ab * ba  as  a ^2 b ^2

Statement (1):
 a  is divisible by 5.
 a  = 5 * integer
 a ^2 = 5^2 * integer^2
Therefore, we can conclude that  a  is a multiple of 25 (amongst other factors, which we do not know).
 a ^2 = 25, 50, 75, 100, ...
Since we do not have information on what  b  is, this alone is insufficient.

Statement (2):
 b  is divisible by 2.
 b  = 2 * integer
 b ^2 = 2^2 * integer^2
Therefore, we can conclude that  b  is a multiple of 4 (amongst other factors, which we do not know).
 b ^2 = 4, 8, 12, 16, ...
Since we do not have information on what  a  is, this alone is insufficient.

(1)+(2):
 a ^2 b ^2 = 5^2 * 2^2 * integer = 25 * 4 * integer = 100 * integer.
Hence, the tens-digit will always be 0 since  a ^2 b ^2 is a multiple of 100.
Hence, sufficient. Answer C.

If a and b are positive integers, What is the value of tens digit of a

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If a and b are positive integers, What is the value of tens digit of a

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