Question

What is the smallest number which when divided by $24,36$ and $54$ gives a remainder of $5$ each time?

Answer 1

Hint: Here, we will first find the least common multiple, which is the smallest positive number that is a multiple of two or more numbers. Then we will add the remainder left by dividing the given divisor from the obtained least common multiple to find the required value.

Complete step-by-step solution:
Here, we have to find the least number which when divided by $24,36$ and $54$ leaves $5$ as the remainder in each case.
Before solving this question, we must know what the division theorem is. Division theorem states that “if ‘n’ is an integer and ‘d’ is a positive integer, there exist unique integers ‘q’ and ‘r’ such that.
$n = dq + r$ where $0 \leqslant r < d$ .
Here, ‘n’ is the number or dividend , ‘d’ is the divisor. ’q’ is the quotient and ‘r’ is the remainder.
We will find the least common multiple of the given numbers $24,36$ and $54$ .
$2\left| \!{\underline {\,
  {24,36,54} \,}} \right. $
$3\left| \!{\underline {\,
  {12,18,27} \,}} \right. $
$3\left| \!{\underline {\,
  {4,6,9} \,}} \right. $
$2\left| \!{\underline {\,
  {4,2,3} \,}} \right. $
$2\left| \!{\underline {\,
  {2,1,3} \,}} \right. $
\[3\left| \!{\underline {\,
  {1,1,3} \,}} \right. \]
   $1,1,1$
We will find the product of the above multiplies to find the least common multiple.
$ \Rightarrow L.C.M. = 2 \times 2 \times 2 \times 3 \times 3 \times 3$
On multiplying the terms, we get
$ \Rightarrow L.C.M. = 216$
Since we get $5$ as the remainder from all the numbers, we will now find the required number.
Adding the number $5$ from the obtained least common multiple, we get
$ = 216 + 5$
$ = 221$
Hence, the smallest number which when divided by $24,36$ and $54$ gives a remainder of $5$ each time is $221$.

Note: Here, the student should first understand what is asked in the question before solving the question. After that, we should figure out that if a number $n$ leaves remainder $r$ when divided by a number $q$ then $n + q - r$ will be a multiple of $q$ and we should apply this concept correctly. Moreover, we should find L.C.M. correctly and calculate the required number.