Question

What is the largest $3$ digit multiple of $7$ ?

Answer 1

Hint: Here, we are asked to calculate the largest three-digit multiple of the number $7$. We know that the largest three-digit number is $999$. So, we shall start with $999$. When we divide the largest digit by seven, we need to get the remainder zero. If we didn’t get zero, we need to proceed by using the next largest-three digit number.

Complete answer:
We know that the largest three-digit number is $999$.
Now, we need to find the largest three-digit multiple of $7$.
a) First, we shall divide the largest three-digit number $999$ by $7$.
$7\mathop{\left){\vphantom{1
  999
  \underline 7
  29
  \underline {28}
  019
  \underline {14}
  05
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
  999
  \underline 7
  29
  \underline {28}
  019
  \underline {14}
  05
 }}}
\limits^{\displaystyle \,\,\, {142}}$
When we divide the largest three-digit number by $7$, we got the remainder $5$ .
But we need the remainder zero. Since we didn’t get the remainder zero, $999$ cannot be the largest $3$ digit multiple of $7$
b) Next, we shall consider the largest three-digit number $998$ .
Now, we shall divide the largest three-digit number $998$ by $7$.
$7\mathop{\left){\vphantom{1
  998
  \underline 7
  29
  \underline {28}
  018
  \underline {14}
  04
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
  998
  \underline 7
  29
  \underline {28}
  018
  \underline {14}
  04
 }}}
\limits^{\displaystyle \,\,\, {142}}$
When we divide the largest three-digit number by $7$, we got the remainder $4$ .
But we need the remainder zero. Since we didn’t get the remainder zero, $998$ cannot be the largest $3$ digit multiple of $7$
c) Next, we shall consider the largest three-digit number $997$ .
Now, we shall divide the largest three-digit number $997$ by $7$.
$7\mathop{\left){\vphantom{1
  997
  \underline 7
  29
  \underline {28}
  017
  \underline {14}
  03
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
  997
  \underline 7
  29
  \underline {28}
  017
  \underline {14}
  03
 }}}
\limits^{\displaystyle \,\,\, {142}}$
When we divide the largest three-digit number by $7$, we got the remainder $3$ .
But we need the remainder zero. Since we didn’t get the remainder zero, $997$ cannot be the largest $3$ digit multiple of $7$
d) Next, we shall consider the largest three-digit number $996$ .
Now, we shall divide the largest three-digit number $996$ by $7$.
$7\mathop{\left){\vphantom{1
  996
  \underline 7
  29
  \underline {28}
  016
  \underline {14}
  02
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
  996
  \underline 7
  29
  \underline {28}
  016
  \underline {14}
  02
 }}}
\limits^{\displaystyle \,\,\, {142}}$
When we divide the largest three-digit number by $7$, we got the remainder $2$ .
But we need the remainder zero. Since we didn’t get the remainder zero, $996$ cannot be the largest $3$ digit multiple of $7$
e) Next, we shall consider the largest three-digit number $995$ .
Now, we shall divide the largest three-digit number $995$ by $7$.
$7\mathop{\left){\vphantom{1
  995
  \underline 7
  29
  \underline {28}
  015
  \underline {14}
  01
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
  995
  \underline 7
  29
  \underline {28}
  015
  \underline {14}
  01
 }}}
\limits^{\displaystyle \,\,\, {142}}$
When we divide the largest three-digit number by $7$, we got the remainder $1$ .
But we need the remainder zero. Since we didn’t get the remainder zero, $995$ cannot be the largest $3$ digit multiple of $7$
f) Next, we shall consider the largest three-digit number $994$ .
Now, we shall divide the largest three-digit number $994$ by $7$.
$7\mathop{\left){\vphantom{1
  994
  \underline 7
  29
  \underline {28}
  014
  \underline {14}
  0
 }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{
  994
  \underline 7
  29
  \underline {28}
  014
  \underline {14}
  0
 }}}
\limits^{\displaystyle \,\,\, {142}}$
When we divide the largest three-digit number by $7$, we got the remainder $0$.
Since we get the remainder zero, $994$ is the largest $3$ digit multiple of $7$

Note:
We can also calculate the required answer by using another method. Here, let us consider the largest-three digit number$999$. When we divide the largest three-digit number by $7$, we got the remainder $5$ . We shall subtract $999$ and $5$
Thus, $999 - 5 = 994$ . Hence we got the required answer.