Question 8

Explanation

Given that $a = 2 \times b$ and $b$ is divisible by $5$, where $a$ and $b$ are natural numbers.

Statement $\text{(1)}$: $a$ is an even number

Since $a = 2b$, $a$ is the result of multiplying any number by $2$. Any number multiplied by $2$ will always result in an even number. Therefore, statement $\text{(1)}$ is true.

Statement $\text{(2)}$: The unit digit of $a$ is always $0$

Since $b$ is divisible by $5$, $b$ can be $5, 10, 15, 20, \ldots$.

• If $b = 5$, then $a = 2 \times 5 = 10$ (unit digit $0$)

• If $b = 10$, then $a = 2 \times 10 = 20$ (unit digit $0$)

• If $b = 15$, then $a = 2 \times 15 = 30$ (unit digit $0$)

Since $b$ is always a multiple of $5$, $2b$ is always a multiple of $10$, so the unit digit of $a$ is always $0$. Therefore, statement $\text{(2)}$ is true.

Statement $\text{(3)}$: $a + b$ is divisible by $5$

Since $b$ is divisible by $5$, $3b$ is also divisible by $5$. Therefore, statement $\text{(3)}$ is true.

Statement $\text{(4)}$: $3a + 2b$ is an odd number

Since $8$ is an even number, $8b$ is always an even number, not an odd number. Therefore, statement $\text{(4)}$ is false.

Therefore, the true statements are $\text{(1)}$, $\text{(2)}$, and $\text{(3)}$. The number of true statements is $3$.