Consider the sets T(n)={n, n+3, n+5, n+7, n+9}
where n=1,2,3,.....,99. How many of these sets contain 5 or any integral multiple there of (i.e. any one of the numbers 5, 10, 15, 20, 25, ...) ?

Options

A. 81
B. 79
C. 80
D. none

Answer

To solve the problem, we need to determine how many sets \( T(n) = \{n, n+3, n+5, n+7, n+9\} \) contain at least one number that is either 5 or an integral multiple of 5 (i.e., 5, 10, 15, 20, ...).

### Step-by-Step Solution:

1. **Understanding the Set \( T(n) \)**:
Each set \( T(n) \) consists of five consecutive odd or even numbers starting from \( n \). The elements of the set are:
\[ T(n) = \{n, n+3, n+5, n+7, n+9\} \]

2. **Finding Multiples of 5**:
We need to find values of \( n \) such that at least one of the elements in \( T(n) \) is a multiple of 5. The multiples of 5 up to 99 are:
\[ 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95 \]
This gives us a total of 19 multiples of 5.

3. **Identifying Valid \( n \)**:
To ensure that at least one element in \( T(n) \) is a multiple of 5, we can express \( n \) in terms of its remainder when divided by 5:
- If \( n \equiv 0 \mod 5 \), then \( n \) is a multiple of 5.
- If \( n \equiv 1 \mod 5 \), then \( n+4 \) is a multiple of 5.
- If \( n \equiv 2 \mod 5 \), then \( n+3 \) is a multiple of 5.
- If \( n \equiv 3 \mod 5 \), then \( n+2 \) is a multiple of 5.
- If \( n \equiv 4 \mod 5 \), then \( n+1 \) is a multiple of 5.

Thus, for any \( n \), at least one of the numbers in \( T(n) \) will be a multiple of 5 if \( n \equiv 0, 1, 2, 3, \) or \( 4 \mod 5 \).

4. **Counting Valid \( n \)**:
Since \( n \) can take values from 1 to 99, we can calculate how many values of \( n \) do not lead to a multiple of 5 in \( T(n) \). The only case where none of the numbers in \( T(n) \) is a multiple of 5 is when \( n \equiv 0 \mod 5 \).

The multiples of 5 from 1 to 99 are:
\[ 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95 \]
This gives us 19 values of \( n \) that do not work.

5. **Calculating Total Sets**:
The total number of values \( n \) can take is 99. Therefore, the number of sets \( T(n) \) that contain at least one multiple of 5 is:
\[ 99 - 19 = 80 \]

### Final Answer:
Thus, the number of sets \( T(n) \) that contain 5 or any integral multiple of 5 is **80**.