P and Q are Prime Numbers Less than 70. What is the Units Digit of P*Q?

Question: P and Q are prime numbers less than 70. What is the units digit of P*Q?

(1) Units’ digit of (\(P^{4k+2}\) - Q) is equal to 7, where k is a positive integer.
(2) Units digit of the expression [PQ+Q*(Q+1) - Q2] is a perfect cube

(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.

“P and Q are Prime Numbers Less than 70. What is the Units Digit of P*Q?” – is a topic of theGMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Approach 1

Given:

• We are given two prime numbers
• P,Q<70

Find Out:

• We are asked to find the units digit of P∗Q

Now, we will be interpreting the problem statement

The unit’s digit of the product of P & Q would depend on the units digit of P & Q
Hence, our mail focus would be to find the units digit of P & Q.

We will check each statement one by one:

Statement I:

The first statement tells us that the difference between two numbers is odd. This would imply opposite even/odd natures of the numbers i.e. one is even and the other is odd. Since, there is only on even number (i.e. 2), there are two possible scenarios:

Let us consider that P=2:

So, P\(^{4k+2}\) will have the units digit as 4
As a result the units digit of Q would be 7
Hence units digit of product of P & Q would be 4

Let us consider that Q=2:
In such a case units digit of P\(^{4k+2}\) will be 9. This implies that units digit of P is either 7 or 3.
Hence, units digit of product of
P & Q would be either 4 or 6.

We are unable to get a definite answer.
Since we do not have a unique answer, Statement-I is not sufficient to answer the question.

Statement II:

The expression in statement-II can be simplified to
Q(P+1), since its unit digit is a perfect cube, the possible values are 1 and 8.

If unit digit is 1, we get:

• Units digit of P= 2 and units digit of Q =7, so the units digit of product of P & Q would be 4.

If unit digit is 8, we get two possible caes:

• Units digit of Q=2 and units digit of P=3, so the units digit of product of P & Q would be 6
• Units digit of Q=1 and units digit of P=7, so the units digit of product of P & Q would be 7

Since we do not have a unique answer, Statement-II is not sufficient to answer the question.

Now, we will combine both statements.

Statement-I tells us that the units digit of
P & Q can be 4 or 6.
Statement-II tells us that the units digit of P & Q can be 4 or 6 or 7.

By Combining statements 1 and 2, we still do not have a unique answer.

Thus combination of statement 1 and statement 2 is also insufficient to answer the question.

Hence, E is the correct option.

Approach 2

Statement 1- Units’ digit of (P4k+2−Q) is equal to 7, where k is a positive integer.

Diff of P&Q is odd. So either of them should be even. So only prime which is even is 2. So Q-2 and To get the diff. 7, P should be 3 as power of 3 in the form of (4k+2) results the unit digit into 9 and 9-2=7.
So P=3 and Q=2.
Hence, the statement is Sufficient.

Statement 2- Units digit of the expression [PQ+Q∗(Q+1)−Q2] is a perfect cube.
Expression modified to (PQ+Q)…
The only unit digit which is perfect cube is 8.

Now, we have Q(P+1)=x8, where x be any number before units digit.
Let us try just 2 -3 prime numbers.
Consider, P=3 Q=2 Q(P+1)=8
PQ=6
Consider, P=3 Q=7 Q(P+1)=28
PQ=21
So, we cannot find the units digit with this condition.
Hence, it is Insufficient making E the correct answer.

Approach 3

Statement 1:
Le’st k=1 then P^6-Q unit digit to be 7 we have two values of P and applying Q same in both the cases
So P=13 and Q=2 gives the unit digit 7 and P=23 and Q=2 will also give 7 as a unit digit
Hence, it is not sufficient

Statement 2:
P*Q-Q unit digit is a perfect cube so the possible value for the perfect cube are 1,8
It is insufficient as there will be two possible values of P and Q
Combining 1 and 2 also we do not get the exact value.

Hence, E is the correct answer.

Correct Answer: E

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