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Steps 1 & 2: Understand Question and Draw InferencesGiven: Two positive integers are given - \(556^{17n}\) and \(339^{5m+15n}\)

To find: The units digit of the sum of these two integers

Analysis: To find units digit let’s concentrate only on the last digit of the number.

Therefore, the expression given to us can be written as

\(6^{17n}\) + \(9^{5m+15n}\)

As we know the units digit of \(6^x\) is always 6, irrespective of the exponent of 6, the units digit of \(6^{17n}\) is 6

The cyclicity of 9 is 2,

If exponent of 9 is 2m, we get the units digit as 1

If the exponent of 9 is 2m+1, the units digit is 9

We need to find the units digit of \(9^{5(m+3n)}\). In order to solve the question we need to find

the value of \(m + 3n \)

OR

B. infer

whether m+3n is odd or evenHence,

if \(m+3n\) is even, then, \(9^{5(m+3n)}\) will be \(9^{even}\) and unit’s digit, in this case will be 1

If \(m+3n\) is odd, then \(9^{5(m+3n)}\) will be \(9^{odd}\) and the unit’s digit, in this case will be 9

Now, let us look at each of the statements.

Step 3: Analyze Statement 1 independently\(4m + 12n = 360\)

\(4(m + 3n) = 360\)

\(m + 3n = 90 = even\)

Hence, we can conclude that \(9^{5(m+3n)} = 9^{even}\)

Therefore, the units digit of \(9^{5(m+3n)}\) is 1

And the unit digit of \(556^{17n}+339^{5m+15n}\) is \(6 + 1 = 7\)

Therefore, Statement 1 is sufficient to get us a unique answer.Step 4: Analyze Statement 2 independentlyStatement 2 says that n is the smallest two-digit number divisible by 5

We can infer that the value of \(n = 10\)

But nothing can be inferred about \(m + 3n\), as we don’t have any information about the value of m.

Therefore, statement 2 is not sufficient to get us a unique answer.Step 5: Analyze Both Statements Together (if needed)Since we’ve already got a unique answer in Step 4, this step is not required.

Correct Answer: Option A
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