To find the integers from 0 to 50 that have a remainder of 1 when divided by 3, we first identify the numbers that satisfy this condition. An integer, n, gives a remainder of 1 when divided by 3 if it can be expressed in the form:
- n = 3k + 1
where k is an integer. Now, let’s determine the feasible values for k when n is between 0 and 50 inclusive.
First, we set up the inequality:
- 0 ≤ 3k + 1 ≤ 50
We can rearrange this to find k:
- -1 ≤ 3k ≤ 49
Now, dividing the entire inequality by 3 gives:
- -1/3 ≤ k ≤ 49/3
Since k must be an integer, the values of k that satisfy this are:
- k = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13
Now, let’s calculate the corresponding values of n for each k:
- For k = 0, n = 1
- For k = 1, n = 4
- For k = 2, n = 7
- For k = 3, n = 10
- For k = 4, n = 13
- For k = 5, n = 16
- For k = 6, n = 19
- For k = 7, n = 22
- For k = 8, n = 25
- For k = 9, n = 28
- For k = 10, n = 31
- For k = 11, n = 34
- For k = 12, n = 37
- For k = 13, n = 40
- For k = 14, n = 43
- For k = 15, n = 46
- For k = 16, n = 49
All of these integers: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, and 49 indeed yield a remainder of 1 when divided by 3. Counting these gives us a total of 17 integers.
Thus, the answer is 17.