Question

Find the least number which when divided by 20, 18 and 30 gives the same remainder 9 in each case.
A. 179
B. 189
C. 169
D. 199

Answer 1

Hint:Here, we will first find the least common multiple (i.e. LCM) of the given three numbers who have the same remainder i.e. 9. As we know, LCM is the smallest positive number that is a multiple of two or more numbers. First, we will find the LCM of all of them and then we will add the remainder given to get the final output.

Complete step by step answer:
Given that, we have to find the least number which when divided by 20, 18 and 30 gives 9 as the remainder in each case. Now, before solving this question, we must know the division theorem. The 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 the dividend, \[d\] is the divisor, \[q\] is the quotient and \[r\] is the remainder.

Next, we will find the LCM of the given numbers i.e. 20, 18, 30.
\[2\left| \!{\underline {\,
  {20,18,30} \,}} \right. \]
\[2\left| \!{\underline {\,
  {10,9,15} \,}} \right. \]
\[3\left| \!{\underline {\,
  {5,9,15} \,}} \right. \]
\[3\left| \!{\underline {\,
  {5,3,5} \,}} \right. \]
\[5\left| \!{\underline {\,
  {5,1,5} \,}} \right. \]
  \[1,1,1\]

Thus, the LCM will be the product of all the above multiplies and so we will have
\[ \Rightarrow 2 \times 2 \times 3 \times 3 \times 5\]
On multiplying the terms, we will get,
\[ \Rightarrow 180\]
Since, we are given 9 as the remainder from all the numbers, we will now find the required number adding the number 9 to the above number to the obtained least common multiple.
So, the required number = (LCM of 20, 18, 30) + remainder
Substituting the values, we will get,
\[ \Rightarrow 180 + 9\]
On evaluating this, we will get
\[ \Rightarrow 189\]

Hence, the least number which when divided by 20, 18 and 30 gives a remainder of 9 every time is 189.

Note:The Least Common Multiple (LCM) is also referred to as the Lowest Common Multiple (LCM) and Least Common Divisor (LCD). Also, the student should first understand what is asked in the question before solving the question and should find LCM correctly and calculate the required number correctly. Then, 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.