Question

A number when divided by 18 leaves a remainder of 15. Which of the following could be the remainder when it is divided by 72?
(A) 33
(B) 51
(C) 15
(D) All of the above

Answer 1

To find the required remainder when it is divided by 72, use the expression ${\text{Dividend}} = {\text{Divisor}} \times {\text{Quotient}} + {\text{Remainder}}$ . We know that when a number is divided by 4, the remainder of the division is 0, 1, 2 or 3. Using these theories we can find the required remainder.

Complete step by step solution:
It is given that the remainder of a number which is divided by 18 is 15.
Consider that the required number be n.
Also consider that x is also a natural number.
It is known that ${\text{Dividend}} = {\text{Divisor}} \times {\text{Quotient}} + {\text{Remainder}}$
According to the question,
$n = 18x + 15$
In the above expression, n is the dividend and x is the quotient.
Also, the number should be divided by 72.
So, rewrite the number $n = 18x + 15$
after dividing it by 72 as,
$\dfrac{n} {{72}} = \dfrac{{18x}} {{72}} + \dfrac{{15}} {{72}}$
Simplify the above number.
$\dfrac{n} {{72}} = \dfrac{{18x}} {{72}} + \dfrac{{15}} {{72}} \\\ \dfrac{n} {{72}} = 4x + \dfrac{{15}} {{72}} \\\$
Write the remainder of $\dfrac{n} {{72}}$ as,
${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15$

We know that the remainder of a number which is divided by 4 is 0, 1, 2, or 3.
Find the required remainder by substituting the values of $\dfrac{x} {4}$ in the expression ${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15$.
Substitute 0 for $\dfrac{x} {4}$
in the expression ${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15$.
${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15 \\\ = 18\left( 0 \right) + 15 \\\ = 0 + 15 \\\ = 15 \\\$
Substitute 1 for $\dfrac{x} {4}$ in the expression ${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15$.
${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15 \\\ = 18\left( 1 \right) + 15 \\\ = 18 + 15 \\\ = 33 \\\$
Substitute 2 for $\dfrac{x} {4}$
in the expression ${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15$.
${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15 \\\ = 18\left( 2 \right) + 15 \\\ = 36 + 15 \\\ = 51 \\\$
Substitute 3 for $\dfrac{x} {4}$ in the expression ${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15$.
${\text{Remainder}}\;{\text{of}}\;\dfrac{n} {{72}} = 18\dfrac{x} {4} + 15 \\\ = 18\left( 3 \right) + 15 \\\ = 54 + 15 \\\ = 69 \\\$
Thus, it is seen that the required remainder when a number is divided by 72 are 15, 33, 51 or 69.
This means the required correct answer is the all the remainders given in the options.
Hence, the required correct answer is (D).

Note:
In order to find the required remainder, it is necessary to find the number which is divided by 18. From this number we can find the remainder of the number when it is divided by 18. Since, there 4 remainders of a number when it is divided by 4, so we can obtain four remainders when divided by 72.