Understanding the Remainder When Dividing

The question asks for the remainder when the sum \(5^{70} + 7^{70}\) is divided by 74. This is a problem involving modular arithmetic, where we need to find the value of \(5^{70} + 7^{70} \pmod{74}\).

Analyzing Bases and Modulus for Remainder

Let's look closely at the numbers involved: the bases are 5 and 7, and the modulus is 74.

Consider the sum of the squares of the bases:

\(5^2 + 7^2 = 25 + 49 = 74\)

Notice that the sum of the squares, \(5^2 + 7^2\), is exactly equal to the modulus, 74. This means \(5^2 + 7^2 \equiv 0 \pmod{74}\).

Using Algebraic Properties for Remainder of Powers

We need to evaluate the expression \(5^{70} + 7^{70}\). We can rewrite the exponents using the square terms we just analyzed:

\(5^{70} = (5^2)^{35}\)

\(7^{70} = (7^2)^{35}\)

So, the original expression becomes \((5^2)^{35} + (7^2)^{35}\).

This expression is in the form \(x^n + y^n\), where:

• \(x = 5^2\)
• \(y = 7^2\)
• \(n = 35\)

The exponent \(n = 35\) is an odd integer.

Applying Divisibility Rule for Remainder

There is a powerful algebraic property related to the sum of powers that is useful in remainder problems:

Property: For any positive odd integer \(n\), the expression \(x^n + y^n\) is always divisible by \(x+y\).

In our specific problem, we have the expression \((5^2)^{35} + (7^2)^{35}\), which perfectly fits the form \(x^n + y^n\) with \(x=5^2\), \(y=7^2\), and \(n=35\) (which is indeed an odd number).

According to this property, the expression \((5^2)^{35} + (7^2)^{35}\) must be divisible by the sum \(5^2 + 7^2\).

We calculated earlier that \(5^2 + 7^2 = 25 + 49 = 74\).

Therefore, this means that \(5^{70} + 7^{70}\) is divisible by 74.

Determining the Final Remainder

By definition, if a number is divisible by another number, the result of the division leaves no remainder. In other words, the remainder is 0.

Since we have established that \(5^{70} + 7^{70}\) is divisible by 74, the remainder when \(5^{70} + 7^{70}\) is divided by 74 is 0.

Final Remainder Calculation Conclusion

Based on the divisibility property of powers, the remainder is 0.

Revision Table: Key Concepts for Remainder Problems

• Modular Arithmetic: The system of arithmetic concerned with remainders after division. Finding the remainder when A is divided by B is equivalent to calculating A mod B.
• Divisibility Properties of Powers: Specific rules like \(x^n + y^n\) being divisible by \(x+y\) for odd \(n\) help simplify calculations involving large exponents.
• Relationship between Divisibility and Remainder: If a number 'a' is divisible by a number 'b', then the remainder when 'a' is divided by 'b' is always 0.

Additional Information: Properties for Remainder Calculations

Understanding the divisibility rules for sums and differences of powers is crucial for efficiently solving many remainder problems. Recall these key properties:

• \(x^n - y^n\) is always divisible by \(x-y\) for any positive integer \(n\). Example: \(x^2 - y^2 = (x-y)(x+y)\), \(x^3 - y^3 = (x-y)(x^2+xy+y^2)\).
• \(x^n - y^n\) is always divisible by \(x+y\) for any positive even integer \(n\). Example: \(x^2 - y^2 = (x-y)(x+y)\), \(x^4 - y^4 = (x^2-y^2)(x^2+y^2) = (x-y)(x+y)(x^2+y^2)\).
• \(x^n + y^n\) is always divisible by \(x+y\) for any positive odd integer \(n\). Example: \(x^3 + y^3 = (x+y)(x^2-xy+y^2)\), \(x^5 + y^5 = (x+y)(x^4-x^3y+x^2y^2-xy^3+y^4)\).

These properties allow us to determine divisibility, and thus the remainder (which is 0 if divisible), without calculating the large powers themselves.