Question

The number of ways to choose 7 distinct natural numbers from the first 100 natural numbers such that any two chosen numbers differ atleast by 7 can be expressed as nC7. Find the number of divisors of n.

• A. The number of divisors of n is 7.
• B. The number of divisors of n is 6.
• C. The number of divisors of n is 8.
• D. The number of divisors of n is 5.

Answer 1

Answer: (A)

The number of divisors of n is 7.

Answer 2

Let the 7 distinct natural numbers chosen from $\{1, 2, \dots, 100\}$ be $x_1, x_2, \dots, x_7$, sorted in ascending order: $1 \le x_1 < x_2 < \dots < x_7 \le 100$. The condition is that $x_{i+1} - x_i \ge 7$ for $i = 1, 2, \dots, 6$.

We use a transformation to simplify this condition. Let $y_1 = x_1$, $y_2 = x_2 - 6$, $y_3 = x_3 - 12$, ..., $y_7 = x_7 - 36$. In general, $y_i = x_i - 6(i-1)$ for $i=1, \dots, 7$.

Now, let's check the differences between consecutive $y_i$: $y_{i+1} - y_i = (x_{i+1} - 6i) - (x_i - 6(i-1)) = x_{i+1} - x_i - 6i + 6i - 6 = (x_{i+1} - x_i) - 6$. Since $x_{i+1} - x_i \ge 7$, we have $y_{i+1} - y_i \ge 7 - 6 = 1$. This means $y_{i+1} > y_i$, so the $y_i$ are distinct and in increasing order: $y_1 < y_2 < \dots < y_7$.

Now, let's find the range of $y_i$. The minimum value of $y_1$ is $x_1 \ge 1$, so $y_1 \ge 1$. The maximum value of $y_7$ is determined by the maximum value of $x_7$: $y_7 = x_7 - 6(7-1) = x_7 - 36$. Since $x_7 \le 100$, the maximum value of $y_7$ is $100 - 36 = 64$.

So, we are choosing 7 distinct numbers $y_1, y_2, \dots, y_7$ from the set $\{1, 2, \dots, 64\}$. The number of ways to do this is $\binom{64}{7}$.

The problem states that this number of ways is equal to $nC7$. Therefore, $\binom{n}{7} = \binom{64}{7}$, which implies $n = 64$.

We need to find the number of divisors of $n=64$. The prime factorization of 64 is $2^6$. The number of divisors of a number $p^a$ is $(a+1)$. So, the number of divisors of $64 = 2^6$ is $(6+1) = 7$. The divisors are $2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6$, which are $1, 2, 4, 8, 16, 32, 64$. There are 7 divisors.