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x and y are integers less than 60 such that x is equal to the sum of the squares of two distinct prime numbers, and y is a multiple of 17. Which of the following could be the value of x – y?
- a)-19
- b)-7
- c)0
- d)4
- e)9
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
x and y are integers less than 60 such that x is equal to the sum of t...
First, let's consider the condition for x. We know that x is equal to the sum of the squares of two distinct prime numbers. The prime numbers less than 60 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, and 59.
We need to find two distinct prime numbers, square them, and add them together to get x.
Now, let's consider the condition for y. It is mentioned that y is a multiple of 17, which means y can be any integer multiple of 17.
Analyzing the options:
A: -19
B: -7
C: 0
D: 4
E: 9
B: -7
C: 0
D: 4
E: 9
To evaluate x - y, we need to check if any of the given options can be obtained by subtracting a possible value of y from x.
Let's analyze each option:
A: x - y = (sum of squares of two primes) - (-19)
B: x - y = (sum of squares of two primes) - (-7)
C: x - y = (sum of squares of two primes) - 0
D: x - y = (sum of squares of two primes) - 4
E: x - y = (sum of squares of two primes) - 9
B: x - y = (sum of squares of two primes) - (-7)
C: x - y = (sum of squares of two primes) - 0
D: x - y = (sum of squares of two primes) - 4
E: x - y = (sum of squares of two primes) - 9
Since y can be any multiple of 17, it will not affect the possibility of x - y being a specific value. Therefore, we only need to focus on the possible values of x.
By analyzing all the prime numbers less than 60, we find that the only possible values for x (sum of squares of two primes) that can be obtained are:
x = 22 + 32 = 4 + 9 = 13
x = 22 + 52 = 4 + 25 = 29
x = 22 + 72 = 4 + 49 = 53
x = 32 + 72 = 9 + 49 = 58
x = 22 + 52 = 4 + 25 = 29
x = 22 + 72 = 4 + 49 = 53
x = 32 + 72 = 9 + 49 = 58
Out of these possible values of x, let's evaluate x - y for each option:
A: 13 - (-19) = 32 (not equal to C)
B: 29 - (-7) = 36 (not equal to C)
C: 13 - 0 = 13 (equal to C)
D: 53 - 4 = 49 (not equal to C)
E: 29 - 9 = 20 (not equal to C)
B: 29 - (-7) = 36 (not equal to C)
C: 13 - 0 = 13 (equal to C)
D: 53 - 4 = 49 (not equal to C)
E: 29 - 9 = 20 (not equal to C)
Therefore, the only option for which x - y could be equal is option C.
Hence, the answer is C.
Most Upvoted Answer
x and y are integers less than 60 such that x is equal to the sum of t...
First, let's consider the condition for x. We know that x is equal to the sum of the squares of two distinct prime numbers. The prime numbers less than 60 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, and 59.
We need to find two distinct prime numbers, square them, and add them together to get x.
Now, let's consider the condition for y. It is mentioned that y is a multiple of 17, which means y can be any integer multiple of 17.
Analyzing the options:
A: -19
B: -7
C: 0
D: 4
E: 9
B: -7
C: 0
D: 4
E: 9
To evaluate x - y, we need to check if any of the given options can be obtained by subtracting a possible value of y from x.
Let's analyze each option:
A: x - y = (sum of squares of two primes) - (-19)
B: x - y = (sum of squares of two primes) - (-7)
C: x - y = (sum of squares of two primes) - 0
D: x - y = (sum of squares of two primes) - 4
E: x - y = (sum of squares of two primes) - 9
B: x - y = (sum of squares of two primes) - (-7)
C: x - y = (sum of squares of two primes) - 0
D: x - y = (sum of squares of two primes) - 4
E: x - y = (sum of squares of two primes) - 9
Since y can be any multiple of 17, it will not affect the possibility of x - y being a specific value. Therefore, we only need to focus on the possible values of x.
By analyzing all the prime numbers less than 60, we find that the only possible values for x (sum of squares of two primes) that can be obtained are:
x = 22 + 32 = 4 + 9 = 13
x = 22 + 52 = 4 + 25 = 29
x = 22 + 72 = 4 + 49 = 53
x = 32 + 72 = 9 + 49 = 58
x = 22 + 52 = 4 + 25 = 29
x = 22 + 72 = 4 + 49 = 53
x = 32 + 72 = 9 + 49 = 58
Out of these possible values of x, let's evaluate x - y for each option:
A: 13 - (-19) = 32 (not equal to C)
B: 29 - (-7) = 36 (not equal to C)
C: 13 - 0 = 13 (equal to C)
D: 53 - 4 = 49 (not equal to C)
E: 29 - 9 = 20 (not equal to C)
B: 29 - (-7) = 36 (not equal to C)
C: 13 - 0 = 13 (equal to C)
D: 53 - 4 = 49 (not equal to C)
E: 29 - 9 = 20 (not equal to C)
Therefore, the only option for which x - y could be equal is option C.
Hence, the answer is C.
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Community Answer
x and y are integers less than 60 such that x is equal to the sum of t...
Analysis:
Given that x is the sum of the squares of two distinct prime numbers and y is a multiple of 17, we need to find the possible values of x - y.
Finding x:
The prime numbers less than 60 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59.
The possible pairs of distinct prime numbers that can be squared and added to get x are (2, 3), (2, 5), (2, 7), (2, 11), (3, 5), (3, 7), (3, 11), (5, 7), (5, 11), (7, 11).
Calculating the sum of squares for each pair gives us the possible values for x: 13, 29, 53, 125, 34, 58, 130, 74, 146, 130.
Finding y:
Multiples of 17 less than 60 are 17, 34, 51.
Therefore, the possible values of y are 17, 34, 51.
Calculating x - y:
Subtracting each possible value of y from the corresponding value of x, we get:
- x - y = 13 - 17 = -4
- x - y = 29 - 34 = -5
- x - y = 53 - 51 = 2
- x - y = 125 - 17 = 108
- x - y = 34 - 34 = 0
- x - y = 58 - 51 = 7
- x - y = 130 - 17 = 113
- x - y = 74 - 34 = 40
- x - y = 146 - 51 = 95
- x - y = 130 - 34 = 96
Therefore, the only possible value of x - y among the given options is 0, which is option C.
Given that x is the sum of the squares of two distinct prime numbers and y is a multiple of 17, we need to find the possible values of x - y.
Finding x:
The prime numbers less than 60 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59.
The possible pairs of distinct prime numbers that can be squared and added to get x are (2, 3), (2, 5), (2, 7), (2, 11), (3, 5), (3, 7), (3, 11), (5, 7), (5, 11), (7, 11).
Calculating the sum of squares for each pair gives us the possible values for x: 13, 29, 53, 125, 34, 58, 130, 74, 146, 130.
Finding y:
Multiples of 17 less than 60 are 17, 34, 51.
Therefore, the possible values of y are 17, 34, 51.
Calculating x - y:
Subtracting each possible value of y from the corresponding value of x, we get:
- x - y = 13 - 17 = -4
- x - y = 29 - 34 = -5
- x - y = 53 - 51 = 2
- x - y = 125 - 17 = 108
- x - y = 34 - 34 = 0
- x - y = 58 - 51 = 7
- x - y = 130 - 17 = 113
- x - y = 74 - 34 = 40
- x - y = 146 - 51 = 95
- x - y = 130 - 34 = 96
Therefore, the only possible value of x - y among the given options is 0, which is option C.
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