e-GMAT Question:

If \(X\) is a positive integer, what is the units digit of \(X^a\)?

1) \(X\) when divided by \(10\) gives a remainder of \(2\)
2) \(8^a\) ends with \(6\)

A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are NOT sufficient

This is

Question 5 of The e-GMAT Number Properties Marathon

Go to

Question 6 of the Marathon

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Solution:

Step 1: Analyse Statement 1:
\(X\), when divided by \(10\), gives a remainder of \(2\)

• Per our conceptual understanding, when a number is divided by \(10\), the remainder is equal to the units digit of the number.

o Here, when X is divided by \(10\), the remainder is \(2\).
o Hence, \(2\) is the units digit of X.
• But to calculate the units digit of the expression, \(X^a\), we need more information on the value of a.

Since we do not have that information,
Statement 1 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choices A and D.
Step 2: Analyse Statement 2:
\(8^a\) ends with \(6\)

• Per our conceptual understanding, we know that the cyclicity of \(8\) is \(4\) and for \(8^a\) end with \(6\), \(a\) should be of the form :\(4k\). From this, we know the nature of the number \(a\), but we do not have any information about\(X\).

Since we do not know the value of \(X\),
Statement 2 alone is NOT sufficient to answer the question.
Hence, we can eliminate answer choice B.

Step 3: Combine both Statements:

• From the first statement, we know: The units digit of \(X\) is \(2\)
• From the second statement we have: a is a multiple of \(4\)

• We know that a is of form 4k, and we also know\(2^{4k}\) always has a units digit: \(6\).

Thus, we know that the units digit of the expression \(X^a\) is \(6\).
By combining both statements we got a unique answer.
Correct Answer: Option C
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