MATH REVOLUTION OFFICIAL SOLUTION:

Since we have 3 variables (n, a, b) and 1 equation (\(n=a^3b^4\)) in the original condition, we need 2 equations to match the number of variables and the number of equations. Since we need both 1) and 2), the correct answer is likely C.

Using con 1) & 2), we get \(n=5^37^4\) or \(n=7^35^4\). The number of different factors is (3+1)(4+1)=20. This is unique and sufficient. Therefore, the correct answer is C.

However, since this is an “integer” question, which is one of the key questions, we should apply Common Mistake Type 4(A).

In case of con 1), if a=b, \(n=a^7\) → (7+1)=8. However, if a≠b, \(n=a^3b^4\) → (3+1)(4+1)=20. So this is not unique and sufficient.

In case of con 2), \(n=5^37^4\) → (3+1)(4+1)=20. However, \(n=1^335^4=5^47^4\) → (4+1)(4+1)=25. This is not unique and sufficient. Therefore the correct answer is C.

Note : For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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If a and b are positive integers, let n=a3∗b4, how many different factors n has?

(1) a and b are prime numbers
(2) n has only prime factors 5 and 7

the number of different factors n will have depends on the prime factorization of a and b ; and the resultant powers of all the prime factors of a and b together. it follows that we need to know the powers of all different prime factors to calculate the total number of factors . we can then apply the formula of (x+1)(y+1)(z+1).. so on where x , y and z are the powers of distinct prime factors that constitute n.

1) says that a and b are prime factors themselves - therefore subsequent factorization is not possible for both a and b. so the number of total factors for n = (3+1)(4+1) - sufficient.
2) says n has only prime factors 5 and 7 . it could very well be the case that a is a multiple of 5 and 7 (each raised to any power) and so is the case with b. it could also be the case that a is 5 raised to power of any positive integer and b is 7 raised to thepower of any positive integer. (or vice versa)
the fact that the product of a^3 and b^7 could be ANY power of 5 or 7 raised subsequently by power of 3 and 7 generates multiple possible powers to both 5 and 7. so a unique set of powers for prime numbers 5 and 7 is not possible

e.g.,
one possibility : a = 5 ; b = 7 : n = 5^3 * 7^4 and the # of factors = (3+1)*(4+1)
another possibility : a = 125 ; b = 49 ; n = 5^9 * 7^8 and the # of factors = (9+1)*(8+1)

both the values are clearly different without needing further calculations - B is clearly insufficient.

Correct Answer : A

We are asked whether we can calculate the number of different factors of \(n\) where \(n = a^3∗b^4\).

Statement 1) tells us that a and b are prime numbers. If \(a\) and \(b\) are prime numbers then their only factors are 1 and themselves, therefore they have exactly 1 factor which is not 1. This lets us calculate the number of factors by calculating the number of different combinations of \(a\) and \(b\) and adding 1 (or including \(a^0 * b^0\) as a combination). Statement 1) is therefore sufficient to determine the number of different factors of n.

Statement 2) states that the only prime factors of \(n\) are 5 and 7. This would mean that the only prime factors of \(a\) and \(b\) are also 5 and 7, but not that \(a\) and \(b\) are necessarily prime. We could therefore have any number of different factors for \(n\), as \(a\) and \(b\) can have any number of different factors themselves. For example 5, 7, 25, 35, 49, 125 could be factors of \(a\) and/or \(b\) and all of these factors would be factors of \(n\) and would be in addition to the number of different combinations of \(a\) and \(b\) as calculated for statement 1). Statement 2) is therefore insufficient as we cannot calculate the number of factors of \(n\) based on its information.

Statement 1) is sufficient and statement 2) is insufficient therefore the answer is A.

If a and b are positive integers, let n=a3∗b4, how many different factors n has?

(1) a and b are prime numbers
(2) n has only prime factors 5 and 7


ANSWER:

n=a^3*b^4

(1) a and b are prime numbers --> number of different factors (including 1 and n) \(= (3+1) * (4+1) = 20\) --> Sufficient
(2) n has only prime factors 5 and 7 --> \(a=5, b=7\) or vice versa --> \(n = 5^1*7^1\) (order is not the issue) --> number of different factors (including 1 and n) \(= (1+1) * (1+1) = 4\) --> sufficient

Answer D

If a and b are positive integers, let n=a 3 ∗b 4 , how many different factors n has?

(1) a and b are prime numbers
(2) n has only prime factors 5 and 7

Explanation:-
1) case a)If a and b are different primes then total no of factors is 20.
case b)if and b are same prime then total no of factors is 9
Hence ,a alone is not sufficient.

2)if 5 and 7 are in a alone or b alone then we can't determine the no of factors.
for multiple of 5 or 7 lso we can't determine the no of factors
Hence ,b alone is not sufficient.
Combining a & b we find a=5,b=7 or b=5 ,a=7.

Hence both a and b together are sufficient. (C)

Find the no. of factors of n given \(n = a^3 * b^4\)

I. a and b are prime => n is in its factorized form.
Therefore total no. of factors are (3+1)(4+1)
Sufficient

II. 5 and 7 are the only factors of 5

but, we don't know that a and b are 5 and 7

It could be that
\(n = 5^3 * 7^4\)
or
\(n = 5^6 * 7^8 = 25^3 * 49^4\)
Not Sufficient

Answer is A

QUESTION #10:
If a and b are positive integers, let n=a3∗b4, how many different factors n has?

(1) a and b are prime numbers
(2) n has only prime factors 5 and 7
Solution:
Statement (1): The number of factors of n can be expressed by the formula = (p+1)(q+1). hereby, for all prime values of a and b , the power remained the same.
N=(3+1)(4+1)=20 . So n has 20 number of factors.
.............Sufficient
Statement (2): By putting power 3 and 4 we can find different factors alongside 5 and 7.
Hereby, statement (2) is not sufficient.

Answer: (A)

If a and b are positive integers, let \(n=a^3∗b^4\), how many different factors n has?

If a and b are prime then we can compute the different factors using their powers. So the question asked is: are a and b are prime numbers?

(1) a and b are prime numbers : sufficient, there are 20 different factors
(2) n has only prime factors 5 and 7: insufficient, knowing that 5 and 7 are the only prime factors does not tell us that they are a and b. a and b could be any other non prime numbers
Answer A
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If a and b are positive integers, let \(n = a^3 * b^4\) , how many different factors n has?

So we are asked to find the total number of factors of \(n\) .


Statement 1: \(a\) and \(b\) are prime numbers.

Since a and b are prime numbers and \(n = a^3 * b^4\) , this means that the expression \(a^3 * b^4\) is the prime factorization of \(n\).
From this, we can find the total number of factors by taking each exponent (here 3 and 4) and adding 1 to it (so we now have 4 and 5) and multiplying these two numbers together.The result (20) is the total number of factors of n.

Note that we don't actually need to go through these steps and arrive at a value. As soon as we know that it is possible to do so, we should move on to the next statement.

Also, note that we don't need to know which prime numbers a and b are as long as we know that \(n = a^3 * b^4\) because it means that \(a\) and \(b\) are the only two prime factors in the prime factorization of \(n\).

Sufficient.


Statement 2 : \(n\) has only prime factors 5 and 7.
\(5\) and \(7\) make up the prime factorization of \(n\) but don't know which exponent to apply to \(5\) and \(7\) so we cannot use the method described above.
Moreover we don't know the value of a so a could be equal to \(5\) or \(7\) or even \(5^2\) or 7^2. The same goes for b. So without the individual values of \(a\) and \(b\) here we cannot determine the number of factors of \(n\).

Not Sufficient


Answer : A

Each statement alone is sufficient.

We are to find the number of factors of n which is given as a^3*b^4.
Now for this we either need to find the value of a and b, or we need to find if a and b are prime numbers.

Statement 1- this gives us exactly what we need. if both a and b are prime no. of factors = (3+1)(4+1) = 20. => sufficient
Statement 2 - this says that the only prime factors n has are 5 & 7. Now this is possible only when either of a or b equals 5 or 7. The actual values doesnt mater as we only need to calculate no of factors. So, no of factors = (3+1)(4+1) = 20. => sufficient.

Thus, answer is each statement alone is sufficient. => option D.