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What is the units digit of a^36? a) a^2 has 9 as the u

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by varun289 » Sat Dec 15, 2012 3:07 am
What is the units digit of a^36?
a) a^2 has 9 as the units digit
b) a^3 has 3 as the units digit


ans - D , while my answer is B , how ?
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by aman88 » Sun Dec 16, 2012 10:32 am
IMO E

Stmt 1: a^2 has 9 as units. According to this, a can of different values such as 7, 13, 17 etc. because we do not know whether a^2 is a 2-digit no. or a 3-digit no. So INSUFFICIENT

Stmt 2: Same as above. There are multiple values for a. So INSUFFICIENT.

Combined: The mystery remains. A can be 17 or 13.
Therefore E.
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by puneetkhurana2000 » Sun Dec 16, 2012 3:16 pm
Statement 1) a^36 = (a^2)^18 Now we know the units digit of a^2 is 9, so Units digit of a^36 = units digit of 9^18 = 1. Sufficient!!

Statement 2) a^36 = (a^3)^12 Now we know the units digit of a^3 is 3, so Units digit of a^36 = units digit of 3^12 = 1. Sufficient!!

Answer D.
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by ceilidh.erickson » Sat Dec 22, 2012 2:17 pm
The value of a as a whole is not necessarily relevant here. All we need to do is establish the units digit of a^36.

Whenever you see a units digit (UD) question involving exponents, the first thing to do is to establish the pattern of the UDs. For example, the pattern of UDs with powers of 3 is as follows:
3^1 = 3
3^2 = 9
3^3 = 7 (only worry about the UD)
3^4 = 1
3^5 = 3
3^6 = 9

You can see that the UD of powers of 3 repeat every 4 exponents. All UDs have similar patterns.

Statement (1) tells us that the UD of a^2 is 9. aman88 is right that we don't know what the UD of a is - it could be 3 or 7, both of which would have a UD of 9 when squared. But that doesn't mean that we don't know what the UD of a^36 is! For both 3 and 7, the UDs cycle every 4 exponents, so 3^4 has a UD of 1, and 7^4 has a UD of 1. So if the exponent is a multiple of 4, and we know that a is either 3 or 7, the UD has to be 1 in either case. The UD of a^36 = 1. SUFFICIENT

Statement (2) tells us that the UD of a^3 is 3. Unlike the first statement, this actually narrows a down to a single units digit: it has to be a UD of 7. If we know the UD of a, we can certainly find the UD of a^36. SUFFICIENT
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
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by lunarpower » Sat Dec 22, 2012 7:16 pm
varun289 wrote:What is the units digit of a^36?
a) a^2 has 9 as the units digit
b) a^3 has 3 as the units digit


ans - D , while my answer is B , how ?
for this particular problem, there's also the following shorter approach:

* for statement 1, note that a^36 is (a^2)^18. therefore, according to the given information, a^36 is (something ending with 9)^18. whatever this is, it's going to be a unique value, so, sufficient.

* for statement 2, note that a^36 is (a^3)^12. therefore, according to the given information, a^36 is (something ending with 3)^12. whatever this is, it's going to be a unique value, so, sufficient.

so, done, (d).

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of course, the approach given by ceilidh (above) is going to be more useful in the long run than this one, because it will apply to more cases.
still, more approaches = better.
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