How many positive integers less than 100 have a reminder of 3 when divided by 7?
13
To find the number of positive integers less than 100 that have a remainder of 3 when divided by 7, we can use the concept of modular arithmetic.
First, let's find the smallest positive integer that has a remainder of 3 when divided by 7. We can do this by finding the smallest multiple of 7 greater than 3. Dividing 3 by 7 gives a quotient of 0 and a remainder of 3. So, the smallest positive integer that satisfies the given condition is 3.
Next, let's find the largest positive integer that satisfies the given condition. The largest multiple of 7 less than 100 is 98. Dividing 98 by 7 gives a quotient of 14 and a remainder of 0. Hence, the largest positive integer that satisfies the given condition is 98 - 7 = 91.
Now, we have to count the number of positive integers between 3 and 91 (inclusive) that have a remainder of 3 when divided by 7. To do this, we can subtract the number of multiples of 7 from the total count of integers.
The range from 3 to 91 contains 89 integers. To find the number of multiples of 7 in this range, we can subtract the highest multiple (91) from the lowest multiple (7) and divide by 7, and then add 1 (to include the highest multiple itself). This gives us (91 - 7) / 7 + 1 = 13 + 1 = 14 multiples of 7.
Therefore, there are 89 - 14 = 75 positive integers less than 100 that have a remainder of 3 when divided by 7.
To determine how many positive integers less than 100 have a remainder of 3 when divided by 7, we can follow these steps:
1. Find the smallest positive integer that has a remainder of 3 when divided by 7. Let's call it x.
- To find x, we need to start with 3 and keep adding 7 until we find a number less than 100.
- In this case, x = 3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87, 94.
2. Find the largest positive integer that has a remainder of 3 when divided by 7. Let's call it y.
- To find y, we need to start with 100 and subtract 7 until we find a number that has a remainder of 3 when divided by 7.
- In this case, y = 94, 87, 80, 73, 66, 59, 52, 45, 38, 31, 24, 17, 10, 3.
3. Calculate the number of positive integers between x and y (inclusive).
- To calculate the number of integers between x and y, we subtract x from y and add 1.
- In this case, the number of integers between x and y is 14.
Therefore, there are 14 positive integers less than 100 that have a remainder of 3 when divided by 7.