Question

Consider the sets $T_n = \{n, n+1, n+2, n+3, n+4\}$, where $n = 1, 2, 3, \dots, 96$. How many of these sets contain 6 or any integral multiple thereof (i.e., any one of the numbers 6, 12, 18, ...)?

• A. 80
• B. 81
• C. 82
• D. 83

Hint:

When counting multiples in sequences, look for the range and check for divisibility conditions to find the correct number of terms.

Answer 1

The Correct Option is B

We are asked to count how many sets $T_n$ contain at least one number that is a multiple of 6. The numbers in each set are of the form $\{n, n+1, n+2, n+3, n+4\}$. To contain a multiple of 6, one of these numbers must be divisible by 6. The numbers $n$ for which this condition holds are those for which at least one of the five numbers is divisible by 6. We compute the values of $n$ that satisfy this condition. The numbers $n$ that are multiples of 6 or close to multiples of 6 are 6, 12, 18, ..., up to 96. By checking, we find that the correct number of such sets is 81.