Study the given statements carefully. State $T$ for true and $F$ for false and select the correct option.
$(i)$ If a number is a factor of each of the given two numbers, then it must be a factor of their difference.
$(i i)$ If a number is divisible by another number, then it must be divisible by each of the factors of that number.
$(i i i)$ If a number is divisible by another number, then it is also divisible by all the multiples of that number.
$(i v)$ No prime number other than $2$ is even but every odd number is necessarily a prime number.

\begin{matrix} (i) & (i i) & (i i i) & (i v) T & F & T & F \end{matrix}

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\begin{matrix} (i) & (i i) & (i i i) & (i v) T & T & F & F \end{matrix}

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\begin{matrix} (i) & (i i) & (i i i) & (i v) F & T & F & T \end{matrix}

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\begin{matrix} (i) & (i i) & (i i i) & (i v) F & F & F & F \end{matrix}

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Solution

The correct option is B $\begin{matrix} (i) & (i i) & (i i i) & (i v) T & T & F & F \end{matrix}$
i) Consider a number $a$ . It is a factor of $k_{1} a$ and $k_{2} a$ , $k_{1} \neq k_{2}, k_{1}, k_{2} \in Z$
Difference between the numbers= $(k_{1} - k_{2}) a$ which is also divisible by $a$ . So statement i is true.
ii) Let the numbers be $a, b$ . If a is a factor of b, then $b = k a, k \in Z$
all the factors, $x_{i}$ of $a$ are of the form, $\frac{a}{k_{i}}$ so $a = k_{i} \times x_{i}, b = k \times k_{i} \times x_{i}$ There fore all the factors of $a$ are also factors of $b$ . So, statement ii is true.
iii)Let the numbers be $a, b$ . If $a$ is a factor of $b$ , then $b = k a, k \in Z$
all the multiples, $x_{i}$ of $a$ are of the form, $a k_{i}$ so $a = \frac{x_{i}}{k_{i}}$
$b = k \times \frac{x_{i}}{k_{i}}$
Therefore all the multiples of $a$ might not be factors of $b$ . So, statement iii is false.
iv) All even numbers greater than 2 are divisible by 2. But all odd numbers are not prime numbers. Ex: 9,15 etc. So, statement iv is False.

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Similar questions

Which of the following statements are true?

(a) If a number is divisible by 3, it must be divisible by 9.

(b) If a number is divisible by 9, it must be divisible by 3.

(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.

(e) If two numbers are co-primes, at least one of them must be prime.

(f) All numbers which are divisible by 4 must also be divisible by 8.

(g) All numbers which are divisible by 8 must also be divisible by 4.

(h) If a number exactly divides two numbers separately, it must exactly divide their sum.

(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.

Question 1

Which of the following statements are true:
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) If a number is divisible by 18, it must be divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are co-primes, at least one of them must be prime.
(f) All numbers which are divisible by 4 must also be divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.

Q. Write (T) for true and (F) for false against each of the following statements:
(i) If a number is divisible by 4, it must be divisible by 8.
(ii) If a number is divisible by 8, it must be divisible by 4.
(iii) If a number divides the sum of two numbers exactly, it must exactly divide the numbers separately.
(iv) If a number is divisible by both 9 and 10, it must be divisible by 90.
(v) A number is divisible by 18 if it is divisible by both 3 and 6.
(vi) If a number is divisible by 3 and 7, it must be divisible by 21.
(vii) The sum of two consecutive odd numbers is always divisible by 4.
(viii) If a number divides two numbers exactly, it must divide their sum exactly.

Q. Which of the following statements are true?

(i) If a number is divisible by 3, it must be divisible by 9.
(ii) If a number is divisible by 9, it must be divisible by 3.
(iii) If a number is divisible by 4, it must be divisible by 8.
(iv) If a number is divisible by 8, it must be divisible by 4.
(v) A number is divisible by 18, if it is divisible by both 3 and 6.
(vi) If a number is divisible by both 9 and 10, it must be divisible by 90.
(vii) If a number exactly divides the sum of two numbers, it must exactly divide the numbers separately.
(viii) If a number divides three numbers exactly, it must divide their sum exactly.
(ix) If two numbers are co-prime, at least one of them must be a prime number
(x) The sum of two consecutive odd numbers is always divisible by 4.

Which of the following statements are true ?
(i) If a number is divisible by 3, it must be divisible by 9.
(ii) If a number is divisible by 9, it must be divisible by 3.
(iii) If a number is divisible by 4, it must be divisible by 8.
(iv) If a number is divisible by 8, it must be divisible by 4.
(v) A number is divisible by 18, if it is divisible by both 3 and 6.
(vi) If a number is bivisible by both 9 and 10, ut nyst be divisible by 90.
(vii) If a number exactly divides the sum of two numbers, it must exactly divide the numbers separately.
(viii) If a number divides three numbers exactly, it must divede their sum exactly.
(ix) If two numbers are co-prime , at least one of them must be a prime number.
(x) The sum of two consecutive odd numbers is always divisible by 4.

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