Question: If x is the sum of factorials of the first n positive integers (i.e. for n=3, x=1!+2!+3!), what is the units digit of x?

(1) n is divisible by 4
(2) (n^2+1)/5 is an odd integer

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

“If x is the sum of factorials of the first n positive integers”– is a topic of the GMAT Quantitative reasoning section of the GMAT exam. The GMAT Quant section comprises a sum of 31 questions. Out of these 31 questions, 15 questions belong to the GMAT data sufficiency section. GMAT Data Sufficiency questions include a problem statement that cites two factual statements. The quantitative section of the GMAT exam improves the candidate’s efficiency in solving mathematical problems. This GMAT quantitative section enables the candidate to test their skills in reasoning, quantitative problems, and graph. The candidates can further enhance their quantitive skills by practising more questions from "GMAT Quantitative Review".

Solution and Explanation:

Approach Solution 1:

The problem statement states that x is the sum of factorials of the first n positive integers. That means for the value of n= 3, the value of x = 1! + 2! + 3!.

We are asked to find out the units digit of x

Naturally, the units digit of 1! + 2! + …. + n! can carry only three values.

1. If the value of n is equal to 1, then the units digit is 1.
2. If the value of n is equal to 3, then the units digit is 9.
3. If n is equal to any further value, then the units digit is 3.
   Therefore, we can say:
   If the value of n is equal to 2, then 1! + 2! = 3.
   If the value of n is equal to 4, then 1! + 2! + 3! + 4! = 33
   If n≥4, then the integers after n=4 will end by 0. It will not impact the units digit and it will stay 3.

According to the factual statement of the question, we can determine these three cases.

1. The statement indicates “n is divisible by 4”. This statement suggests that the value of n is not 1 or 3. Thus it satisfies the third case, Therefore the statement alone is sufficient.
2. The statement implies “(n^2+1)/5 is an odd integer”. Therefore, the value of n is not 1 or 3. Thus this statement also supports the third case. Hence, the statement alone is sufficient.

Thus we can say, each of the statements alone is sufficient. Hence, D is the correct answer.

Correct Answer: D

Approach Solution 2:

Any series of multiplication that includes 2*5 will always possess units digit as zero.

We need to calculate the values of n till 4!. After this, the unit digit would be similar since from n=5, all terms from 5! will possess units as zero.

(n=2) => 1 + 2! = 3
(n=3) => 3 + 3! = 9
(n=4) => 9 + 4! = 33

All the terms that need to be added will have units digit as zero since they include 2*5, so units digit will always be 3.

(n=5) => 33 + 5! (include 2*5) = 123
(n=5) =>33+ 6! (include 2*5)= 753 and so on.

Therefore, both statements alone are sufficient. Hence D is the correct answer.

Correct Answer: D

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