Question

Among the numbers $29540,53416\,$ and $21543$, which of the given options is true?
A) None are divisible by 12.
B) One is divisible by 12.
C) Two are divisible by 12.
D) All are divisible by 12.

Answer 1

Hint - To solve this question, firstly we have to understand the properties of division and then by using it as a parameter we can solve this question. Then we will simply check the divisibility by 12 by checking the divisibility of the number by 4 and 3. If yes, then the number is divisible by 12. This will help us in approaching the solution.

Complete step by step solution:
For the numbers $29540\,,53416$ and $21543$ to be divisible by $12$,
they must be divisible by $3$ as well as $4$ (since $3 \times 4 = 12$)
$29,540 - $is divisible by $4$ but not $3$ .... as last two digits $40$ is divisible by $4$ but sum of digits $ = 2 + 9 + 5 + 4 = 20$ ...not divisible by $3$
$53,416 - $ is divisible by $4$ since last two digits are $16$ which is divisible by $4$ but not $3$ because the sum of all the digits i.e. $5 + 3 + 4 + 1 + 6 = 19$ is not divisible by 3.
$21,543 - $ is divisible by $3$ because the sum of all the digits is $2 + 1 + 5 + 4 + 3 = 15$ which is divisible by $3$
 but not $4$ since the last two digits are 43 which is not divisible by 4.
Hence, the three numbers are not divisible by $12$ because none of them is divisible by both 3 and 4.
So, option A is correct. None are divisible by 12.
Also, For a number N to be divisible by a number A, N must be divisible by the multiples of A. i.e. for any number to be divisible by 12, it must be divisible by the multiples of 12 i.e. 3 and 4 and to find this, we can add the numbers and then check the divisibility of the number by 3 which we can check by adding all the digits of that number. For example, 15, 1 + 5 = 6. And 6 is divisible by 3. Hence 15 is divisible by 3. This basic knowledge can be very useful in solving questions of this kind.

Note - Properties of division
If a and b are two whole numbers then a ÷ b is not mandatorily a whole number.
e.g. - Consider the division of 14 by 3. There is no whole number when multiplied by 3 gives us 14. So, 14 ÷ 3 is not a whole number.
1) Let a is any whole number, then a ÷ 1 = a.
e.g. - $5 \div 1 = 5$
2) If a is any whole number other than zero, then a ÷ a = 1.
E.g. -$13 = 13 \times 1$
Therefore,$13 \div 13 = 1$
3) When zero is divided by any whole number except zero, it gives out the quotient as zero. We can also say that, if a is a whole number other than zero, then 0 ÷ a = 0
e.g. - $0 \times 7 = 0$
So, $0 \div 7 = 0$
4) Let a, b and c are three whole numbers where b ≠ 0, c ≠ 0. Suppose, a ÷ b = c, then b × c = a.
e.g.:$15 \div 3 = 5$
So, $5 \times 3 = 15$