49. How many integers between 70 and 105 have a reminder of 2 when divided by 15?
15*5+2 = 77
15*6+2 = 92
So, only two of them
5*15 + 2 = 77
6*15 + 2 = 92
To find the number of integers between 70 and 105 (inclusive) that have a remainder of 2 when divided by 15, we need to determine the range of possible values that satisfy the given condition.
We can start by finding the first number in the range that satisfies the condition. To do this, we need to find the smallest integer greater than or equal to 70 that has a remainder of 2 when divided by 15.
Dividing 70 by 15 gives a quotient of 4 and a remainder of 10. Since 10 is not equal to 2, we need to find the next multiple of 15 that has a remainder of 2. Adding the remainder (2) to 70, we get 72.
Now we have the first number in the range of integers between 70 and 105 that satisfies the condition: 72.
Next, we find the last number in the range. Similar to finding the first number, we need to find the largest integer less than or equal to 105 that has a remainder of 2 when divided by 15.
Dividing 105 by 15 gives a quotient of 7 and a remainder of 0. Since 0 is not equal to 2, we need to find the previous multiple of 15 that has a remainder of 2. Subtracting the remainder (2) from 105, we get 103.
Now we have the last number in the range of integers between 70 and 105 that satisfies the condition: 103.
To find the number of integers in this range, we can subtract the first number (72) from the last number (103) and add 1 to account for the inclusive range.
103 - 72 + 1 = 32
Therefore, there are 32 integers between 70 and 105 (inclusive) that have a remainder of 2 when divided by 15.