What is the remainder when (17 29 + 19 29 ) is divided by 18?

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Prepp Team · Accepted Answer

Calculating Remainder Using Modular Arithmetic and Properties

We are asked to find the remainder when the expression $(17^{29} + 19^{29})$ is divided by 18. This is a problem that can be efficiently solved using the principles of modular arithmetic.

Method 1: Using Modular Arithmetic Properties

The core idea of modular arithmetic is looking at the remainder when a number is divided by another number. We want to find $(17^{29} + 19^{29})$ modulo 18.

Let's consider the base 17 with respect to the divisor 18. $17 \equiv - 118$ This is because 17 is one less than 18.
Now, let's raise this congruence to the power of 29: $17^{29} \equiv {(- 1)}^{29} 18$
Since 29 is an odd exponent, ${(- 1)}^{29}$ is equal to -1. $17^{29} \equiv - 118$
Next, consider the base 19 with respect to the divisor 18. $19 \equiv 118$ This is because 19 is one more than 18.
Now, raise this congruence to the power of 29: $19^{29} \equiv 1^{29} 18$
Since any power of 1 is 1, $1^{29}$ is equal to 1. $19^{29} \equiv 118$
Now, we need to find the remainder of the sum $(17^{29} + 19^{29})$ . We can add the congruences: $17^{29} + 19^{29} \equiv (- 1) + 118$
Simplifying the right side: $17^{29} + 19^{29} \equiv 018$

This means that when $(17^{29} + 19^{29})$ is divided by 18, the remainder is 0.

Method 2: Using Algebraic Identity

There is a useful algebraic identity that states that for any positive odd integer $n$ , the expression $a^{n} + b^{n}$ is always divisible by $(a + b)$ .

In our problem, we have the expression $17^{29} + 19^{29}$ .
Here, $a = 17$ , $b = 19$ , and the exponent $n = 29$ .
Since 29 is an odd number, we can apply the identity.
According to the identity, $17^{29} + 19^{29}$ is divisible by $(a + b)$ , which is $(17 + 19)$ .
Let's calculate the sum $(17 + 19)$ : $17 + 19 = 36$
So, $17^{29} + 19^{29}$ is divisible by 36.
Now, consider the divisor in the question, which is 18. Since 36 is a multiple of 18 ( $36 = 2 \times 18$ ), any number that is divisible by 36 is also divisible by 18.
Therefore, $17^{29} + 19^{29}$ is divisible by 18.

When a number is divisible by another number, the remainder is always 0.

Both methods confirm that the remainder when $(17^{29} + 19^{29})$ is divided by 18 is 0.

Revision Table: Key Concepts for Remainder Problems

Concept	Description	Example (mod 18)
Modular Arithmetic	System of arithmetic for integers, where numbers "wrap around" upon reaching a certain value (the modulus). We look at remainders.	$20 \equiv 2 pmod>18$
Congruence Relation	$a \equiv b pmod>m$ means $a - b$ is divisible by $m$ . $a$ and $b$ have the same remainder when divided by $m$ .	$17 \equiv - 1 pmod>18$ because $17 - (- 1) = 18$ , which is divisible by 18.
Properties of Exponents in Modulo	If $a \equiv b pmod>m$ , then $a^{n} \equiv b^{n} pmod>m$ for any non-negative integer $n$ .	Since $19 \equiv 1 pmod>18$ , $19^{29} \equiv 1^{29} pmod>18$ .
Sum Property of Congruences	If $a \equiv b pmod>m$ and $c \equiv d pmod>m$ , then $a + c \equiv b + d pmod>m$ .	Since $17^{29} \equiv - 1 pmod>18$ and $19^{29} \equiv 1 pmod>18$ , $17^{29} + 19^{29} \equiv - 1 + 1 pmod>18$ .
Divisibility Identity for Odd Powers	For odd $n$ , $a^{n} + b^{n}$ is divisible by $a + b$ .	$17^{29} + 19^{29}$ is divisible by $17 + 19 = 36$ , since 29 is odd.

Additional Information on Remainder Calculations

Calculating remainders, especially with large numbers and exponents, is a common topic in number theory and is often tested in exams. Here are some related concepts:

Fermat's Little Theorem: If $p$ is a prime number, then for any integer $a$ not divisible by $p$ , we have $a^{p - 1} \equiv 1 pmod>p$ . This can simplify calculations with prime moduli.
Euler's Totient Theorem: This is a generalization of Fermat's Little Theorem. If $n$ is a positive integer, then for any integer $a$ coprime to $n$ , we have $a^{φ (n)} \equiv 1 pmod>n$ , where $φ (n)$ is Euler's totient function. This helps with composite moduli like 18.
Chinese Remainder Theorem: Useful for finding a number that has specific remainders when divided by several different numbers.
Negative Remainders: As seen in the solution, sometimes using negative remainders (like -1 mod 18) can simplify calculations significantly. The final positive remainder is obtained by adding the modulus if the result is negative (e.g., $- 1 \equiv - 1 + 18 \equiv 17 pmod>18$ ). In this problem, the sum was 0 mod 18, which is a standard remainder.

Prepp Team · Answer

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Calculating Remainder Using Modular Arithmetic and Properties

Method 1: Using Modular Arithmetic Properties

Method 2: Using Algebraic Identity

Revision Table: Key Concepts for Remainder Problems

Additional Information on Remainder Calculations

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