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How many of the first 1000 positive integers can be expressed in the form [2x] + [4x] + [6x] + [8x], where is a real number, and [z] denotes the greatest integer less than or equal to z ?
Correct answer is '600'. Can you explain this answer?
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How many of the first 1000 positive integers can be expressed in the f...
Approach:
To solve this problem, we can use the principle of inclusion-exclusion. We will count the numbers that can be expressed in each of the forms [2x], [4x], [6x], and [8x], and then subtract the numbers that are counted more than once. Finally, we will add the numbers that are counted thrice.
Solution:
Counting the numbers that can be expressed in each of the forms [2x], [4x], [6x], and [8x]:
- The numbers that can be expressed in the form [2x] are 2, 3, 4, 5, 6, 7, ..., 1998, 1999, 2000.
- The numbers that can be expressed in the form [4x] are 4, 5, 6, 7, 8, 9, ..., 1996, 1997, 1998, 1999, 2000.
- The numbers that can be expressed in the form [6x] are 6, 7, 8, 9, 10, 11, ..., 1994, 1995, 1996, 1997, 1998, 1999, 2000.
- The numbers that can be expressed in the form [8x] are 8, 9, 10, 11, 12, 13, ..., 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000.
Subtracting the numbers that are counted more than once:
- The numbers that can be expressed in the forms [2x] and [4x] are 4, 5, 6, 7, 8, 9, ..., 1996, 1997, 1998, 1999, 2000.
- The numbers that can be expressed in the forms [2x] and [6x] are 6, 7, 8, 9, 10, 11, ..., 1994, 1995, 1996, 1997, 1998, 1999, 2000.
- The numbers that can be expressed in the forms [2x] and [8x] are 8, 9, 10, 11, 12, 13, ..., 1992, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000.
- The numbers that can be expressed in the forms [4x] and [6x] are 6, 7, 8, 9, 10, 11, ..., 1994, 1995, 1996, 1997, 1998, 1999, 2000.
- The numbers that can be expressed in the forms [4x] and [8x] are 8, 9, 10, 11, 12
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How many of the first 1000 positive integers can be expressed in the form [2x] + [4x] + [6x] + [8x], where is a real number, and [z] denotes the greatest integer less than or equal to z ?Correct answer is '600'. Can you explain this answer?
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