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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.
In remainder questions, you get the same answer if you do the divisions first and calculate with the remainders or do the calculations first and then find the remainder.
Since 12 = 13*1 + (-1), 12, the remainder when 12^{mn} is divided by 13 is the same as the remainder when (-1)^{mn} is divided by 13.
If mn is an odd number, (-1)^{mn} = -1 and if mn is an even number, (-1)^{mn} = 1.
The question asks if mn is an odd number or an even number.

Condition 1)
If m is an even integer, mn is an even number and the remainder when 12^{mn} is divided by 13 is 1.
Condition 1) is sufficient, since it yields a unique answer.

Condition 2)
If m = 2 and n = 1, then 12^{2*1} = 12^2 = 144 = 13*11 + 1 and the remainder is 1.
If m = 1 and n = 1, then 12^{1*1} = 12^1 = 12 = 13*0 + 12 and the remainder is 12.
Condition 2) is not sufficient since it does not yield a unique answer.

Therefore, A is the answer.
Answer: A

Statement 1==> m = even
This means that n is either odd or even 1 and the product m will always be even
$$12^{mn}$$
where mn=even number
If mn=4, then
$$12^4=\frac{20736}{13}=1595\ remainder\ 1$$
If mn=6, then
$$12^6=\frac{2985984}{13}=229691\ remainder\ 1$$
If mn=8, then
$$12^8=\frac{429981696}{13}=33075515\ remainder\ 1$$
$$Thus\ \frac{12^{mn}}{13},\ the\ remainder\ will\ always\ =1\ when\ m=even\ and\ product\ of\ \left(mn\right)=even$$
Statement 1 is SUFFICIENT

Statement 2==> n = odd
This means that m is either even or odd, so the product mn will either be (even x odd) or (odd x odd).
Hence, mn = even or mn = odd

$$For\ 12^{mn}\ when\ mn=even,\ the\ remainder\ =1$$
$$For\ 12^{mn}\ when\ mn=odd$$
$$If\ mn=3,\ then\ 12^3=\frac{1728}{13}=132,\ remainder\ is\ 12$$
$$If\ mn=5,\ then\ 12^5=\frac{248832}{13}=19140,\ remainder\ is\ 12$$
$$If\ mn=7,\ then\ 12^7=\frac{35831808}{13}=2756292,\ remainder\ is\ 12$$
$$But,\ when\ mn=even,\ \frac{12^{mn}}{13}\ will\ its\ remainder\ equal\ to\ 1$$
$$And\ when\ mn=odd,\ \frac{12^{mn}}{13}\ will\ have\ its\ remainder\ equal\ to\ 12$$
$$Therefore,\ there\ is\ no\ specific\ remainder.\ Hence,\ STATEMENT\ 2\ IS\ NOT\ SUFFICIENT$$ <i class="em em-clap"></i>

So,
$$STATEMENT\ 1\ alone\ is\ SUFFICIENT.\ Thereby\ making\ OPTION\ A\ the\ correct\ answer.\ Thanks$$