How Many 4-Digit Perfect Squares Start with the Digit 7?

Understanding the range of 4-digit perfect squares that begin with the digit 7 requires a little bit of mathematical reasoning and arithmetic. This article will guide you through the process of determining how many such perfect squares exist and provide step-by-step solutions.

The Range of 4-Digit Perfect Squares

First, let's establish the range of 4-digit perfect squares. A 4-digit number ranges from 1000 to 9999.

If a perfect square starts with the digit 7, it must lie between 6999 (just below 7000) and 8000 (just below 8000). This means we are looking for numbers in the form of 7XX, where X represents any digit from 0 to 9.

Identifying the Range of Squareroots

To find the perfect squares between 6999 and 8000, we need to find the square roots of 6999 and 8000.

Square root of 6999: Rounding up, we get sqrt(6999) ≈ 84. Square root of 8000: Rounding down, we get sqrt(8000) ≈ 89.

This indicates that the perfect squares we are interested in are those obtained by squaring the integers from 84 to 89 inclusive.

Calculating the Perfect Squares

Let's list out the perfect squares of the numbers from 84 to 89:

842 7056 852 7225 862 7396 872 7569 882 7744 892 7921

As a result, there are six 4-digit perfect squares that start with the digit 7, which are 7056, 7225, 7396, 7569, 7744, and 7921.

Brute Force Solution Using J Programming Language

Using the J programming language, we can efficiently find these perfect squares by filtering through the results:

Code:

s~7{.
7056 7225 7396 7569 7744 7921

This code filters the perfect squares that start with the digit 7, providing the same six results as our manual calculation.

Conclusion

To sum up, there are exactly six 4-digit perfect squares that start with the digit 7. These squares are the result of squaring the integers from 84 to 89. Understanding the range of square roots and performing the calculations step-by-step not only provides a clear answer but also reinforces the importance of mathematical reasoning in solving such problems.