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# The product of the predecessor and the successor of the greatest 2-digit number is A. 9900B. 9800C. 9700D. None of these

Last updated date: 24th Jul 2024
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Hint: We find the greatest 2-digit number to be 99. Then we find the predecessor and the successor of the greatest 2-digit number to be 98 and 100 respectively. We multiply them to find the solution. We also use the identity $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ to find the solution of the multiplication.

Complete step by step solution:
The greatest 2-digit number is 99.
The predecessor and the successor of the greatest 2-digit number will be 98 and 100.
We need to find the product of the predecessor and the successor of the greatest 2-digit number which is equal to $98\times 100=9800$.
We can also use the identity $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$.
We assume $a=99;b=1$. Then we get $\left( 99-1 \right)\left( 99+1 \right)=98\times 100$.
Therefore, the solution would be $\left( 99-1 \right)\left( 99+1 \right)=98\times 100={{99}^{2}}-1=9801-1=9800$.
The correct option is (B).

Note:
We need to remember that the terms predecessor and the successor define the exact previous and next terms. We get them by subtracting 1 and adding 1 respectively.