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01 Nov 2022, 01:30
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If z = x*y^3, where x and y are positive integers, what is the value of y ?
(1) The number of different positive factors of z is 4.
(2) z is the product of two different prime numbers.
(1) The number of different positive factors of z is 4.
(2) z is the product of two different prime numbers.
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01 Nov 2022, 02:50
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Given
\( z = x * y^3\)
\(x \geq{1}\) ; \(y \geq{1}\)
Question
y = ?
Statement 1
The number of different positive factors of z is 4.
\( z = x * y^3\)
Now y can be = 1, and x can be a cube of a prime number.
So \(x = p^3\)
\(z = p^3 * 1\)
Alternatively,
y can be any prime number and x = 1
Hence, we can eliminate A and D
Statement 2
z is the product of two different prime numbers.
So, \(z = p_1 * p_2\)
Also \( z = x * y^3\)
Now there can be multiple scenarios we can think of
Note: We don't even have to consider scenario 1 and scenario 2 (to save time). It's given that y is an integer, so if y has any prime factors in it, z will be a product of more than two prime factors. Hence y has to be equal to 1.
Thus have a definite answer.
IMO B
\( z = x * y^3\)
\(x \geq{1}\) ; \(y \geq{1}\)
Question
y = ?
Statement 1
The number of different positive factors of z is 4.
\( z = x * y^3\)
Now y can be = 1, and x can be a cube of a prime number.
So \(x = p^3\)
\(z = p^3 * 1\)
Alternatively,
y can be any prime number and x = 1
Hence, we can eliminate A and D
Statement 2
z is the product of two different prime numbers.
So, \(z = p_1 * p_2\)
Also \( z = x * y^3\)
Now there can be multiple scenarios we can think of
- y is a prime number and x is also a prime number : In that case, z is a multiple of more than two prime numbers, hence this is not a valid case.
- y is a prime number and x = 1 : In that case, z is again a multiple of more than two prime numbers, hence this is also not a valid case.
- \(x = p_1 * p_2\) and y = 1 : This is valid.
Note: We don't even have to consider scenario 1 and scenario 2 (to save time). It's given that y is an integer, so if y has any prime factors in it, z will be a product of more than two prime factors. Hence y has to be equal to 1.
Thus have a definite answer.
IMO B
01 Nov 2022, 17:05
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given
z = x*y^3, where x and y are positive integers
target value of y?
#1
The number of different positive factors of z is 4.
z=x*y^3
case 1 ; x is 1 and y is prime number
case 2 : x is cube of prime number and y is 1
insufficient
#2
z is the product of two different prime numbers
least value of z is 6
z = x*y^3
x has to be 6 and y is 1
sufficient
option B
z = x*y^3, where x and y are positive integers
target value of y?
#1
The number of different positive factors of z is 4.
z=x*y^3
case 1 ; x is 1 and y is prime number
case 2 : x is cube of prime number and y is 1
insufficient
#2
z is the product of two different prime numbers
least value of z is 6
z = x*y^3
x has to be 6 and y is 1
sufficient
option B
Bunuel
