If p and q are positive integers and X = 6p + 7q+23, what is the units digit of X?(1) q = 2p 11 (2) q2 10q + 9 = 0a)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.Correct answer is option 'B'. Can you explain this answer?

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If p and q are positive integers and X = 6^p + 7^q+23, what is the units digit of X?

(1) q = 2p – 11

(2) q² – 10q + 9 = 0

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.

Correct answer is option 'B'. Can you explain this answer?

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Test: Unit Digit- 1

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If p and q are positive integers and X = 6p + 7q+23, what is the units...

Steps 1 & 2: Understand Question and Draw Inferences

We are given that X = 6^p + 7^q+23, and we have to find the unit digit of X. The numbers p and q both are positive integers. Now, the unit digit of X will be the sum of the unit digits of 6^p and 7^q+23.

So, we have to find the unit digit of 6^p and 7^q+23.

Now, we know that the unit digit of 6 raised to any integer power is 6. So, the unit digit of 6^p is 6.

And the cyclicity of 7 is 4 i.e. every 4th power of 7 has the same unit digit and cycles of power of 7 are 7, 9, 3, and 1.

So, the unit digit of the expression:

X = 6^p + 7^q+23 = (Unit Digit of 6^p) + (Unit digit of 7^q+23)

= 6 + (Unit digit of 7^q+23)

So, the unit digit of 7^q+23 will depend on the value of q. We have to find the value of q to determine the unit digit of X.

Step 3: Analyze Statement 1

Statement 1 says: q = 2p – 11

However, since we don’t know the value of p, we can’t determine the value of q.

Hence, statement I is not sufficient to answer the question: What is the unit digit of X?

Step 4: Analyze Statement 2

Statement 2 says:

q² – 10q + 9 = 0

q² – 9q -q + 9 = 0

(q – 9) (q – 1) = 0

Thus, q= 1, 9

Let’s consider both the values one by one:

If q = 1

In this case, the expression 7^q+23 becomes:

7¹⁺²³= 7²⁴

Since the cyclicity of 7 is 4 i.e. every 4th power of 7 has the same unit digit and cycles of power of 7 are 7, 9, 3, and 1.

And,

24 = 4*6

So, the unit digit of 7²⁴ = 1

2. If q = 9

In this case, the expression 7^q+23 becomes:

79⁺²³= 7³²

Since the cyclicity of 7 is 4 i.e. every 4th power of 7 has the same unit digit and cycles of power of 7 are 7, 9, 3, and 1.

And,

32 = 4*8

So, the unit digit of 7³² = 1

So, in both the cases we get the unit digit of 7^q+23 as 1. As derived in the first step, the unit digit of X

= 6 + (Unit digit of 7^q+23) = 6 + 1 = 7

So, statement (2) alone is sufficient to answer the question: What is the unit digit of X?

Step 5: Analyze Both Statements Together (if needed)

Since statement (2) alone is sufficient to answer the question, we don’t need to perform this step.

Answer: Option (B)

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If p and q are positive integers and X = 6p + 7q+23, what is the units digit of X?(1) q = 2p – 11 (2) q2 – 10q + 9 = 0a)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.Correct answer is option 'B'. Can you explain this answer?

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