Question

Find the least number which when divided by $ 16 $ , $ 18 $ and $ 21 $ leaves the remainder $ 3 $ , $ 5 $ and $ 8 $ respectively.

Answer 1

Hint: Here subtract the numbers with the remainder terms. Then the required number will be equal to the least common factor of the numbers minus $ 13 $ .

Complete step-by-step answer:
The given numbers are $ 16 $ , $ 18 $ and $ 21 $ .
We know that the least number which when divided by $ 16 $ , $ 18 $ and $ 21 $ leaves the remainder terms as $ 3 $ , $ 5 $ and $ 8 $ respectively.
Now, let us subtract number $ 16 $ with its remainder, we get,
 $ 16 - 3 = 13 $
Now let us subtract number $ 18 $ with its remainder, we get
 $ 18 - 5 = 13 $
Now let us subtract number $ 21 $ with its remainder, we get
 $ 21 - 8 = 13 $
Hence the numbers subtracted with respective remainders gives $ 13 $ . Since,
 $ 16 - 3 = 18 - 5 = 21 - 8 = 13 $
Thus , $ 16 $ , $ 18 $ and $ 21 $ leaves a remainder $ 3 $ , $ 5 $ and $ 8 $ is always divided by $ 13 $ .
To find the least number when divided by $ 16 $ , $ 18 $ and $ 21 $ leaves a remainder $ 3 $ , $ 5 $ and $ 8 $ is,
We know that the formula to find the least number is,
 $ {\rm{least number}} = {\rm{LCM}}\left( {16,18,21} \right) - 13 $
Now, let us find the least common factor for $ 16 $ , $ 18 $ and $ 21 $ is,
The multiples for 16 is $ 2 \times 2 \times 2 \times 2 $ .
The multiplier for $ 18 $ is $ 3 \times 3 \times 2 \times 2 $ .
The multiplier for $ 21 $ is $ 7 \times 3 $ .
The least common factor for $ 16 $ , $ 18 $ and $ 21 $ is $ 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 $ .
Hence, the least common factor for $ 16 $ , $ 18 $ and $ 21 $ is $ 1008 $ .
We know the formula to find the least number is given by,
 $ {\rm{least number}} = {\rm{LCM}}\left( {16,18,21} \right) - 13 $
On substituting the least common factor for $ 16 $ , $ 18 $ and $ 21 $ in the above formula we obtain,
 $ \begin{array}{c}
{\rm{least number}} = \dfrac{{{\rm{LCM}}\left( {16,18,21} \right) - 13}}{1}\\
 = 1008 - 13\\
 = 995
\end{array} $
Hence, the least number which when divided by $ 16 $ , $ 18 $ and $ 21 $ leaves a remainder $ 3 $ , $ 5 $ and $ 8 $ is $ 995 $ .


Note: Since, there is a common number that is $ 13 $ . If the terms are not common then we need to take the least divisible term common for all. To find least common factor there is another formula also that is,
 $ {\rm{LCM}}\left( {p,q,r} \right) = \dfrac{{p \cdot q \cdot r \cdot {\rm{HCF}}\left( {p,q,r} \right)}}{{{\rm{HCF}}\left( {p,q} \right) \cdot {\rm{HCF}}\left( {q,r} \right) \cdot {\rm{HCF}}\left( {p,r} \right)}} $