Question

28 divides 37744 exactly. Which two numbers nearest to $ 37744 $ are exactly divisible by $ 28 $ ?

Answer 1

Hint: We can observe that the multipliers of a number $ x $ are always in A.P with the common difference $ d=x $ . In this problem we have given that $ 28 $ divides $ 37744 $ exactly. So the number $ 37744 $ is a term in the A.P of which starts with $ 28 $ and has a common difference as $ 28 $ . Let $ 37744 $ be the $ {{n}^{th}} $ term in the A.P, from this we will calculate the value of $ n $ from the known formula from A.P i.e. $ {{a}_{n}}=a+\left( n-1 \right)d $ . To find the nearest value to $ 37744 $ that can be divisible by $ 28 $ , we will calculate the $ {{\left( n-1 \right)}^{th}} $ term and $ {{\left( n+1 \right)}^{th}} $ term of the A.P.

Complete step by step answer:
Given that,
 $ 28 $ divides $ 37744 $ exactly.
Now the series formed with multipliers of $ 28 $ is given by $ 28,28+1\left( 28 \right),28+2\left( 28 \right),...,37744,... $ .
Let $ 37744 $ be the $ {{n}^{th}} $ term of the above series, then $ {{a}_{n}}=37744 $ . But we have the formula for the $ {{n}^{th}} $ term of the A.P as
 $ \begin{align}
  & {{a}_{n}}=a+\left( n-1 \right)d \\
 & \Rightarrow 37744=28+\left( n-1 \right)28 \\
 & \Rightarrow 37716=\left( n-1 \right)28 \\
 & \Rightarrow n-1=1347 \\
 & \Rightarrow n=1348 \\
\end{align} $
So, we have $ 37744 $ as $ {{1348}^{th}} $ term in the series of multipliers. Now the nearest values to $ 37744 $ that can be divisible by $ 28 $ are the $ {{\left( n-1 \right)}^{th}} $ term and $ {{\left( n+1 \right)}^{th}} $ term of the A.P.
 $ \begin{align}
  & \therefore {{a}_{n-1}}=a+\left( n-1-1 \right)d \\
 & \Rightarrow {{a}_{n-1}}=28+\left( 1348-2 \right)28 \\
 & \Rightarrow {{a}_{n-1}}=28+1346\times 28 \\
 & \Rightarrow {{a}_{n-1}}=37716 \\
\end{align} $ and $ \begin{align}
  & \therefore {{a}_{n+1}}=a+\left( n+1-1 \right)d \\
 & \Rightarrow {{a}_{n+1}}=28+\left( 1348 \right)28 \\
 & \Rightarrow {{a}_{n+1}}=28+37744 \\
 & \Rightarrow {{a}_{n+1}}=37772 \\
\end{align} $
So the nearest values to the number $ 37744 $ and that can be divisible by $ 28 $ are $ 37716 $ , $ 37772 $ .

Note: We can directly find the solution for this problem by simply subtracting and adding $ 28 $ to $ 37744 $. The term we will get when we subtract $ 28 $ from $ 37744 $ is $ 37744-28=37716 $ and the term when we add $ 28 $ to $ 37744 $ is $ 37744+28=37772 $ . From both the methods we got the same result. You can check the result by dividing the obtained results by $ 28 $ to check whether the result is correct or not.