Let us solve each of the given problems step by step.

(iii) Problem

If $d(n)$ denotes the number of positive divisors of $n$, then $d(12) =$ ?

Solution:

To determine $d(12)$, we first find the prime factorization of $12$: $12 = 2^2 \cdot 3^1$

The formula to compute the number of divisors is: $d(n) = (e_1 + 1)(e_2 + 1)\ldots(e_k + 1)$ where $e_1, e_2, \ldots, e_k$ are the exponents of the prime factors.

For $12 = 2^2 \cdot 3^1$, the exponents are $2$ and $1$. So: $d(12) = (2 + 1)(1 + 1) = 3 \cdot 2 = 6$

Thus, $d(12) = 6$.

Answer: $b) \, 6$

(iv) Problem

If $x$ is an integer, then $[x] + [-x] =$ ?

Solution:

Here, $[x]$ represents the greatest integer less than or equal to $x$ (floor function). Let's analyze the behavior:

1. If $x$ is positive: [ [x] = x, \quad [-x] = -x - 1 \quad (\text{since } -x \text{ is negative, and we floor it to the next lower integer}). ] Therefore: [ [x] + [-x] = x + (-x - 1) = -1 ]

2. If $x$ is negative: [ [x] = x, \quad [-x] = -x \quad (\text{since } -x \text{ is positive, and the floor is just } -x). ] Therefore: [ [x] + [-x] = x + (-x) = -1 ]

In both cases, $[x] + [-x] = -1$.

Answer: $d) \, -1$

(v) Problem

If $(a, b) = g$, then a linear Diophantine equation $ax + by = c$ has a solution if and only if:

Solution:

The linear Diophantine equation $ax + by = c$ has a solution if and only if the greatest common divisor ($\text{gcd}$) of $a$ and $b$, which is $g$, divides $c$. That is: $g \mid c$

This is a standard result from number theory.

Answer: $c) \, g \mid c$

Final Answers:

• (iii): $b) \, 6$
• (iv): $d) \, -1$
• (v): $c) \, g \mid c$

Would you like detailed explanations or further clarification on any of these problems?

Related Questions:

1. How do you calculate the number of divisors for any number using its prime factorization?
2. What are the properties of the floor function $[x]$ for different types of numbers?
3. Why does $g \mid c$ ensure that a Diophantine equation has a solution?
4. Can there be multiple solutions to $ax + by = c$? How do you find them?
5. How would the solution to $[x] + [-x]$ change if $x$ were not an integer?

Tip:

The formula for the number of divisors $d(n)$ is powerful and can also help find the sum and product of all divisors when combined with other number-theoretic tools!