NIMCET 2007 — Reasoning PYQ

Step 1: Identify the highest four-digit number

The largest possible four-digit number is:

$$\text{Highest 4-digit number}=9999$$

Step 2: Find the closest perfect square using the long division method

To find what must be subtracted from $9999$ to make it a perfect square, we find its square root using the long division method to determine the remainder.

1. Group the digits in pairs from right to left: $\overline{99}\overline{99}$.

2. Find the largest number whose square is less than or equal to $99$. That number is $9$ (since $9^2=81$).

       9  9
     _______
  9 | 99 99
    |-81
     _______
 189| 18 99
    |-17 01
     _______
        1 98  <-- Remainder

• Detailed Calculation:

  • Subtract $81$ from $99$ to get $18$, then bring down the next pair ($99$) to get $1899$.

  • Double the quotient $9$ to get $18$. Now find a digit $x$ such that $18x\times x\le 1899$.

  • Trying $x=9$:

    $$189\times 9=1701$$

  • Subtract $1701$ from $1899$:

    $$\text{Remainder}=1899-1701=198$$

Step 3: Conclusion

The remainder obtained is $198$. This means that $9999$ is greater than the perfect square $99^2$ by exactly $198$.

$$99^2=9801$$

$$9999-198=9801$$

Therefore, the number that must be subtracted from the highest four-digit number to make it a perfect square is $198$.

Correct Answer

The correct option is (b) 198.