Understanding Remainders with Modular Arithmetic

The question asks us to find the remainder when a specific linear combination of numbers, (4x - 2y + 3z), is divided by 31. We are given the remainders when the individual positive numbers x, y, and z are divided by 31.

This problem can be efficiently solved using the properties of modular arithmetic. Modular arithmetic deals with remainders after division.

Given Information on Remainders

We are given the following information:

• When x is divided by 31, the remainder is 17. This can be written in modular notation as: \(x \equiv 17 \pmod{31}\).
• When y is divided by 31, the remainder is 24. This can be written as: \(y \equiv 24 \pmod{31}\).
• When z is divided by 31, the remainder is 27. This can be written as: \(z \equiv 27 \pmod{31}\).

We need to find the remainder of \((4x - 2y + 3z)\) when divided by 31, which is equivalent to finding \((4x - 2y + 3z) \pmod{31}\).

Applying Modular Arithmetic Properties

A key property of modular arithmetic states that if \(a \equiv b \pmod{m}\) and \(c \equiv d \pmod{m}\), then:

• \(a + c \equiv b + d \pmod{m}\)
• \(a - c \equiv b - d \pmod{m}\)
• \(ka \equiv kb \pmod{m}\) for any integer k.

We can use these properties to find the remainder of the expression \((4x - 2y + 3z)\) modulo 31.

Step-by-Step Calculation of the Remainder

Let's find the remainder for each term in the expression \((4x - 2y + 3z)\) when divided by 31.

1. Find the remainder of \(4x\) when divided by 31:
   Since \(x \equiv 17 \pmod{31}\), we have \(4x \equiv 4 \times 17 \pmod{31}\).
   \(4 \times 17 = 68\).
   Now find the remainder of 68 when divided by 31:
   \(68 = 2 \times 31 + 6\). So, \(68 \equiv 6 \pmod{31}\).
   Therefore, \(4x \equiv 6 \pmod{31}\).
2. Find the remainder of \(-2y\) when divided by 31:
   Since \(y \equiv 24 \pmod{31}\), we have \(-2y \equiv -2 \times 24 \pmod{31}\).
   \(-2 \times 24 = -48\).
   Now find the remainder of -48 when divided by 31. A remainder must be non-negative and less than the divisor (31).
   We can write \(-48 = -2 \times 31 + 14\). So, \(-48 \equiv 14 \pmod{31}\).
   Alternatively, \(-48 \equiv -48 + 31 \pmod{31} \equiv -17 \pmod{31}\), and \(-17 \equiv -17 + 31 \pmod{31} \equiv 14 \pmod{31}\).
   Therefore, \(-2y \equiv 14 \pmod{31}\).
3. Find the remainder of \(3z\) when divided by 31:
   Since \(z \equiv 27 \pmod{31}\), we have \(3z \equiv 3 \times 27 \pmod{31}\).
   \(3 \times 27 = 81\).
   Now find the remainder of 81 when divided by 31:
   \(81 = 2 \times 31 + 19\). So, \(81 \equiv 19 \pmod{31}\).
   Therefore, \(3z \equiv 19 \pmod{31}\).

Now, we can combine the remainders using the addition and subtraction properties of modular arithmetic for the expression \((4x - 2y + 3z)\).

\((4x - 2y + 3z) \equiv (6 + 14 + 19) \pmod{31}\)

Calculate the sum of the remainders:

\(6 + 14 + 19 = 20 + 19 = 39\)

So, \((4x - 2y + 3z) \equiv 39 \pmod{31}\).

Finally, we need to find the remainder of 39 when divided by 31. This is the final remainder for the expression.

\(39 = 1 \times 31 + 8\).

Thus, \(39 \equiv 8 \pmod{31}\).

The remainder when \((4x - 2y + 3z)\) is divided by 31 is 8.

Conclusion

By applying the properties of modular arithmetic, we calculated the remainder for each term and then combined them to find the final remainder of the entire expression when divided by 31. The resulting remainder is 8.

Revision Table: Understanding Remainders

Additional Information on Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus. It is widely used in number theory, cryptography, and computer science.

• The expression \(a \equiv b \pmod{m}\) means that \(a\) and \(b\) have the same remainder when divided by \(m\). Equivalently, \(a - b\) is a multiple of \(m\).
• Working with remainders directly simplifies calculations involving large numbers or expressions, as shown in this problem. Instead of calculating the large number \((4x - 2y + 3z)\) first and then dividing by 31, we worked with the smaller remainders throughout the process.
• When dealing with negative numbers in modular arithmetic, the remainder must be a non-negative integer between 0 and \(m-1\). For example, \(-48 \pmod{31}\) is 14, not -17, because \(14\) is in the range \([0, 30]\).