Question

Find the greatest number of four digits which when divided by 8, 9 and 10 leaves 5 as a remainder in each case.
(a) 9730
(b) 9725
(c) 9715
(d) 9995

Answer 1

Hint: First of all take the L.C.M of 8, 9 and 10 then note it down. After that we are going to find the greatest four digit number. We know that the greatest four digit number is 9999. Then divide 9999 by the L.C.M of 8, 9 and 10. After division you will get some remainder subtract the remainder from 9999. Now, add this result of subtraction to 5 because it is given that the four digit greatest number dividing by 8, 9 and 10 should give remainder 5.

Complete step-by-step answer:
We have to find the greatest four digit number which on division with 8, 9 and 10 leaves 5 as remainder in each case.
We know that the greatest four digit number is 9999.
Now, take the L.C.M of 8, 9 and 10 which we are going to find by first of all writing the prime factorization of 8, 9 and 10.
Prime factorization of 8, 9 and 10 are as follows:
$\begin{align}
  & 8=2\times 2\times 2\times 1 \\
 & 9=3\times 3\times 1 \\
 & 10=2\times 5\times 1 \\
\end{align}$
L.C.M is calculated by multiplying the factors which are common in all the three numbers and then multiply the uncommon factors of all the three numbers.
1 is a factor which is common in all the three numbers.
The multiplications of uncommon factors of the three numbers are as follows:
$\begin{align}
  & 2\times 2\times 2\times 3\times 3\times 5 \\
 & =360 \\
\end{align}$
Now, multiplying common factor 1 with uncommon factors 360 we get,
360.
Hence, L.C.M of 8, 9 and 10 is 360.
Now, dividing 9999 by 360 we get,
$360\overset{27}{\overline{\left){\begin{align}
  & 9999 \\
 & \dfrac{720}{\begin{align}
  & 2799 \\
 & \dfrac{2520}{279} \\
\end{align}} \\
\end{align}}\right.}}$
From the above division, we have found that the remainder is 279. Now, subtracting 279 from 9999 we get,
$\begin{align}
  & 9999-279 \\
 & =9720 \\
\end{align}$
Now, 9720 is the greatest four digit number which on division with 8, 9 and 10 leave remainder 0 but we want the greatest four digit number which on division with 8, 9 and 10 leave remainder 5 so we are going to add 5 in 9720 so that it becomes a greatest four digit number which on division by 8, 9 and 10 will leave 5 as a remainder.
$\begin{align}
  & 9720+5 \\
 & =9725 \\
\end{align}$
Hence, 9725 is the greatest four digit number which on division by 8, 9 and 10 will leave 5 as a remainder in each case.
Hence, the correct option is (b).

Note: The alternative way to solve the above problem is to divide each option by 8, 9 and 10 then see which option on division with these three numbers will give 5 as a remainder.
Checking option (a) 9730,
Dividing 9730 by 8 we get,
$8\overset{1216}{\overline{\left){\begin{align}
  & 9730 \\
 & \dfrac{8}{\begin{align}
  & 17 \\
 & \dfrac{16}{\begin{align}
  & 13 \\
 & \dfrac{8}{\begin{align}
  & 50 \\
 & \dfrac{48}{02} \\
\end{align}} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}$
On dividing 9730 by 8 we are getting remainder as 2. But the question says that the four digit greatest number when divided by 8, 9 and 10 should leave a remainder 5 so this option is incorrect.
Now, checking option (b) 9725,
Dividing 9725 by 8 we get,
$8\overset{1215}{\overline{\left){\begin{align}
  & 9725 \\
 & \dfrac{8}{\begin{align}
  & 17 \\
 & \dfrac{16}{\begin{align}
  & 12 \\
 & \dfrac{8}{\begin{align}
  & 45 \\
 & \dfrac{40}{05} \\
\end{align}} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}$
We are getting the remainder as 5.
Now, dividing 9725 by 9 we get,
$9\overset{18}{\overline{\left){\begin{align}
  & 9725 \\
 & \dfrac{9}{\begin{align}
  & 72 \\
 & \dfrac{72}{005} \\
\end{align}} \\
\end{align}}\right.}}$
In the above, we are getting the remainder as 5.
Now, dividing 9725 by 10 we get,
$10\overset{972}{\overline{\left){\begin{align}
  & 9725 \\
 & \dfrac{90}{\begin{align}
  & 72 \\
 & \dfrac{70}{\begin{align}
  & 25 \\
 & \dfrac {20}{5} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}$
In the above division, we have got the remainder as 5.
Hence, option (b) i.e. 9725 on division by 8, 9 and 10 will leave 5 as remainder in each case.
Hence, the correct option is (b). Similarly, you can check the other options too but this will be a long approach to divide every option, so we do not give priority to this method.