Question

Find the largest number dividing 230 and 142 and leaving remainder 5 and 7 respectively.

Answer 1

Hint: First of all subtract the remainders from their corresponding numbers like subtracting 5 from 230 and subtracting 7 from 142 and write their subtractions. Now, write the prime factorization of the subtracted results then find the H.C.F of the two numbers. H.C.F is the largest number which divides 230 and 142 and leaves remainder 5 and 7 respectively.

Complete step-by-step answer:
We have given the two numbers 230 and 142 which on dividing by the largest number will leave the remainders as 5 and 7 respectively. We are asked to find the largest number.
First of all subtract the remainders from their respective dividends.
Subtracting 5 from 230 and 7 from 142 we get,
$\begin{align}
  & 230-5=225 \\
 & 142-7=135 \\
\end{align}$
Now, find the H.C.F of the above two subtraction results. H.C.F of 225 and 135 is calculated by writing the prime factorization first then the factors which are common in both the prime factorization is the H.C.F.
Prime factorization for 225 is:
$225=\underline{3\times 3\times 5}\times 5$
Prime factorization for 142 is:
$135=\underline{5\times 3\times 3}\times 3$
As you can see that underlined factors are common in both 225 and 135 so the multiplication of underlined factors is the H.C.F.
H.C.F is equal to:
$\begin{align}
  & 5\times 3\times 3 \\
 & 45 \\
\end{align}$
Hence, the largest number is 45 which divides 230 and 142 and leaves remainder 5 and 7 respectively.


Note:You can check whether 45 is the largest number by dividing 230 and 142 and leaving remainder 5 and 7 respectively by dividing 230 with 45 and 142 with 45 and see whether the remainders are the same as given in the question or not.
Dividing 230 by 45 we get,
$45\overset{5}{\overline{\left){\begin{align}
  & 230 \\
 & \dfrac{225}{005} \\
\end{align}}\right.}}$
In the above, you can see that we are getting the remainder as 5 which is the same as given in the question.
Dividing 142 by 45 we get,
$45\overset{3}{\overline{\left){\begin{align}
  & 142 \\
 & \dfrac{135}{007} \\
\end{align}}\right.}}$
In the above, you can see that we are getting the remainder as 7 which is the same as given in the question.
Hence, we have shown that 45 on dividing 230 and 142 and leaving remainder 5 and 7 respectively.