Introduction

When faced with the problem of finding the least number that should be subtracted from a given number to make it a perfect square, the solution often involves a combination of mathematical techniques such as modular arithmetic and prime factorization. This article aims to provide a detailed step-by-step procedure to solve this problem without the need for a calculator.

Identifying the Perfect Square

Let's consider the problem: What least number should be subtracted from 19766 to make the resulting number a perfect square?

We start with the number 19766.

First, we test various numbers near 19766 to see if they can be perfect squares. We begin by checking numbers such as 18227, 18229, and so on, until we find a perfect square. Next, we calculate 18227:

18227 ≡ 3 (mod 4)

This does not satisfy the criteria for a perfect square, as a perfect square must be congruent to 0, 1 (mod 4).

Next, we test 18229, and we find that:

18229 ≡ 5 (mod 8)

This is also not a perfect square as all odd perfect squares are congruent to 1 (mod 8).

Further Analysis and Modular Arithmetic

We continue testing, and find that:

18224 is a candidate:

18224 ≡ 2 (mod 3)

This does not satisfy the criteria as any number ending in 224 when added to 18 (a number divisible by 9) is not divisible by 3, and hence, 18224 is not a perfect square.

18228 also does not satisfy the criteria as 18 is divisible by 9 but 228 is not. 18222 and 18226 are both divisible by 2 but not 4, thus, they are not perfect squares. 18223 is a candidate. Any form of 4k3 (4k 3) is not a perfect square. We test 18225:

Note that 18225's last two digits 25 and the first two digits 18 can provide useful clues.

18225 ≡ 18 (mod 27) and 225 ≡ 9 (mod 27).

189 27, and when we divide 18225 by 27, we get a quotient divisible by 3.

Since 225 is the square of 15 and the last two digits are 25, we can test 18225 against the property of perfect squares.

18225/25 729, a perfect square, 3^6.

Square root of 3^6 27.

Thus, 19766 - 18225 2711. The least number to subtract is 2711 - 25 2.

Prime Factorization of 18225

To find the prime factorization of 18225, we start with its square root, 135:

135 ≡ 5 (mod 5), hence: 135/5 27. 27 is a perfect cube, 27 3^3.

So, 135 3^3 * 5.

Therefore, the prime factorization of 18225 is:

18225 3^6 * 5.

Concluding Remarks

The least number to be subtracted from 19766 to make it a perfect square is 2, as demonstrated through modular arithmetic and prime factorization.