A function is defined as f (n) = the number of factors of n. If f (p*q*r) = 8, where p, q and r are prime numbers, what is the value of p?(1) p + q + r is an even number (2) q p r 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 'E'. Can you explain this answer?

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A function is defined as f (n) = the number of factors of n. If f (p*q*r) = 8, where p, q and r are prime numbers, what is the value of p?

(1) p + q + r is an even number

(2) q < p < r

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 'E'. Can you explain this answer?

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A function is defined as f (n) = the number of factors of n. If f (p*q...

Steps 1 & 2: Understand Question and Draw Inferences

We are given that:

f(n) = the number of factors of n and

f(p*q*r) = 8

This means,

Number of factors of pqr = 8

We have learnt that in order to find the total number of factors of any number X, we should write it in terms of its prime factors:

X = P₁^m * P₂ⁿ* P₃^r . . . where P₁, P₂, P₃ . . . are the prime factors of X and m, n, r . . . are non-negative integers

The total number of prime factors of X = (m+1)(n+1)(r+1) . . .

Here, it is given that:

Total number of factors of pqr = 8

Now, 8 = 2³ = (1+1)(1+1)(1+1)

Thus, total number of factors of pqr = (1+1)(1+1)(1+1)

Comparing this with the above formula, we can deduce that p, q and r are distinct prime numbers.

[Note: We can deduce that p, q and r are distinct prime numbers because if say, p was equal to r, in that case, we could write:

pqr = p²q

The total number of factors in this case would be = (2+1)(1+1) = 3*2 = 6.

Since we are given the total number of factors is not 6, we can say confidently that no two prime numbers are equal to one another]

We know that except 2, all the prime numbers are odd. 2 is the only even prime number.

Step 3: Analyze Statement 1

Statement (1) says:

p + q + r = even number.

Now, the sum of three integers can be an even number only if:

Either all the integers are even numbers. (Think 2 + 4 + 6 = 12)
Or one of them is even and two are odd numbers. (Think 2 + 3 + 5 = 10)

Since p, q, and r are distinct prime numbers, all three of them can’t be even because there is only one even prime number.

So, the only possible condition is that one of p, q, and r is even number and the other two are odd numbers.

So, this statement tells us that the value of one of the numbers p, q, and r is 2. However, it does not tell us what the value of p is.

To conclude

One of p, q, or r is equal to 2.

So, statement (1) alone is not sufficient to answer the question “what is the value of p”.

Step 4: Analyze Statement 2

Statement (2) tells us that q < p < r. So, we know that the prime number p is greater than q and smaller than r.

However, since there are infinite number of prime numbers possible, we still cannot determine the value of p.

To conclude

q < p < r

Thus, statement (2) alone is not sufficient to answer the question “what is the value of p”.

Step 5: Analyze Both Statements Together (if needed)

Since the individual analysis of the choices does not tell us the value of p, let’s analyse both the statements together.

From statement (1): One of p, q, or r is equal to 2.

From statement (2): q < p < r

Now, since 2 is the smallest prime number, the value of q is 2. So, p is a prime number greater than 2, but we still don’t know the value of p.

So, analyzing both the statements together is not enough to determine the value of p.

Answer: Option (E)

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A function is defined as f (n) = the number of factors of n. If f (p*q*r) = 8, where p, q and r are prime numbers, what is the value of p?(1) p + q + r is an even number (2) q < p < r 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 'E'. Can you explain this answer?

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