Question

Find the largest $2$ digit number that divides $673$ and $865$ leaving the remainder $1$ in each.

Answer 1

Hint: Here a very important concept of HCF is used. You can do this question with the help of this concept only and also you should have the knowledge of the remainder. We should keep in mind the number we have to find here is the two-digit number.

Complete step by step solution:
In the given question, we have
First, we will find the numbers which will left after subtracting the given numbers from the remainder.
Therefore,
$ \Rightarrow 673 - 1 = 672$\[\]
$ \Rightarrow 865 - 1 = 864$
Now factorize both the numbers,
$ \Rightarrow 672 = 2 \times 2 \times 2 \times 2 \times \,2 \times 3 \times 7$
$ \Rightarrow 864 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$
Therefore,
The HCF of $672\,\,and\,\,864\, = 2 \times 2 \times 2 \times 2 \times 2 \times 3$
The HCF of $672\,\,and\,\,864\, = 96$
Therefore, the largest two-digit number that divides $673\,\,and\,\,865$ by leaving the remainder $1$ in each is $96.$

Note: The Highest Common Factor (HCF) of two numbers is the highest possible number which divides both the numbers exactly. The highest common factor (HCF) is also called the greatest common divisor (GCD. There are some important points about HCF like: $\left( 1 \right)$ HCF of two numbers divides each of the numbers without leaving any remainder.
$\left( 2 \right)$ HCF of two numbers is a factor of each of the numbers.
$\left( 3 \right)$ HCF of two numbers is always less than or equal to each of the numbers. $\left( 4 \right)$ HCF of two prime numbers is 1 always.