Key Concept: Modular Arithmetic and Properties of Exponents This problem involves finding the remainder of a large expression when divided by 18. The most systematic and efficient method to solve this is by using modular arithmetic, which deals with remainders after division. The core idea is that if two numbers are congruent modulo $m$ (i.e., they have the same remainder when divided by $m$), then we can substitute one for the other within an expression when working modulo $m$. Specifically, for exponents, if $a \equiv b \pmod m$, then $a^n \equiv b^n \pmod m$. We will leverage the cyclic nature of remainders for powers of a number.

Problem Breakdown We need to find the remainder of the expression $3 \times 7^{22} + 2 \times 10^{22} - 44$ when divided by 18. This can be mathematically expressed as finding the value of: $(3 \cdot 7^{22} + 2 \cdot 10^{22} - 44) \pmod{18}$ We will evaluate each term of the expression modulo 18 separately and then combine the results.

Step 1: Evaluate $7^{22} \pmod{18}$ To find the remainder of $7^{22}$ when divided by 18, we examine the pattern of the powers of 7 modulo 18:

• $7^1 \equiv 7 \pmod{18}$
• $7^2 = 49$. Since $49 = 2 \times 18 + 13$, we have $7^2 \equiv 13 \pmod{18}$.
• $7^3 \equiv 7 \times 13 = 91$. Since $91 = 5 \times 18 + 1$, we have $7^3 \equiv 1 \pmod{18}$.

Explanation: We found that $7^3$ leaves a remainder of 1 when divided by 18. This is a powerful result, as it means the remainders repeat in a cycle of length 3 ($7, 13, 1, 7, 13, 1, \dots$). To find $7^{22} \pmod{18}$, we divide the exponent 22 by the cycle length 3: $22 = 3 \times 7 + 1$. Therefore, we can rewrite $7^{22}$ as $(7^3)^7 \times 7^1$. Taking this modulo 18: $7^{22} \equiv (1)^7 \times 7^1 \pmod{18}$ $7^{22} \equiv 1 \times 7 \pmod{18}$ $7^{22} \equiv 7 \pmod{18}$ This means $7^{22}$ leaves a remainder of 7 when divided by 18.

Step 2: Evaluate $10^{22} \pmod{18}$ Next, let's find the remainder of $10^{22}$ when divided by 18:

• $10^1 \equiv 10 \pmod{18}$
• $10^2 = 100$. Since $100 = 5 \times 18 + 10$, we have $10^2 \equiv 10 \pmod{18}$.

Explanation: Notice a special pattern here: $10^1 \equiv 10 \pmod{18}$ and $10^2 \equiv 10 \pmod{18}$. This implies that for any integer exponent $n \ge 1$, $10^n \equiv 10 \pmod{18}$. This is because if $a^k \equiv a \pmod m$, then $a^{k+1} \equiv a \cdot a^k \equiv a \cdot a \equiv a^2 \pmod m$. If $a^2 \equiv a \pmod m$, then $a^{k+1} \equiv a \pmod m$. Therefore, $10^{22} \equiv 10 \pmod{18}$ This means $10^{22}$ leaves a remainder of 10 when divided by 18.

Step 3: Evaluate $-44 \pmod{18}$ For the constant term, we simply find its remainder when divided by 18. Since we want a non-negative remainder: $-44 = -3 \times 18 + 10$ So, $-44 \equiv 10 \pmod{18}$ Explanation: When working with negative numbers in modular arithmetic, we add multiples of the modulus (18 in this case) until the result is a non-negative remainder between 0 and $m-1$.

Step 4: Combine the Results Now, substitute the individual remainders (found in Steps 1, 2, and 3) back into the original expression modulo 18: $(3 \cdot 7^{22} + 2 \cdot 10^{22} - 44) \pmod{18}$ $\equiv (3 \cdot (7) + 2 \cdot (10) - (10)) \pmod{18}$ Perform the multiplications and subtractions: $\equiv (21 + 20 - 10) \pmod{18}$ $\equiv (41 - 10) \pmod{18}$ $\equiv 31 \pmod{18}$

Step 5: Final Remainder Finally, we find the remainder of 31 when divided by 18: $31 = 1 \times 18 + 13$ So, $31 \equiv 13 \pmod{18}$ The remainder is 13.

Discrepancy Note My detailed calculation, based on standard modular arithmetic principles and the most logical interpretation of the question's exponents, yields a remainder of 13. This differs from the provided "Correct Answer" of 3 and the original solution's implied remainder of 15. The original solution appears to make incorrect assumptions that $7^{22} \pmod{18}$ and $10^{22} \pmod{18}$ are both equal to 1, which is not true. It is possible there was a transcription error in the question or the provided correct answer.

Tips and Common Mistakes

• Non-negative Remainders: Always ensure your final remainder is a non-negative integer. If a calculation yields a negative result (e.g., -3), add the modulus repeatedly until it becomes positive (e.g., $-3 + 18 = 15$).
• Cyclicity is Key: For large exponents, systematically finding the cycle of remainders (i.e., finding $a^k \equiv 1 \pmod m$ or a repeating pattern) is crucial for simplifying calculations.
• Careful with $\gcd(a, m) \ne 1$: Euler's Totient Theorem ($a^{\phi(m)} \equiv 1 \pmod m$) only applies when $a$ and $m$ are coprime ($\gcd(a,m)=1$). For $10^{22} \pmod{18}$, since $\gcd(10,18)=2 \ne 1$, we cannot directly use Euler's theorem and must rely on direct observation of the pattern of powers.

Summary and Key Takeaway This problem demonstrates the effectiveness of modular arithmetic for handling expressions with large numbers and exponents. By breaking the problem into smaller parts, calculating congruences for each term, and then combining them, we can systematically determine the overall remainder. It's essential to be meticulous with calculations and understand the specific properties of modular arithmetic for different base numbers and moduli.