What is the remainder when 15²³ + 23²³ is divided by 19?

Introduction / Context:
This is a modular arithmetic question involving large powers. Directly computing 15²³ and 23²³ is impossible in an exam setting, so you must use properties of remainders and congruences. The key ideas are reducing bases modulo 19 and exploiting patterns in powers, especially when numbers are negatives of each other modulo the divisor. Such problems test deeper understanding of number theory techniques.

Given Data / Assumptions:

• Expression: 15²³ + 23²³.
• Divisor: 19.
• We seek the remainder when the expression is divided by 19.
• All calculations are done using integer arithmetic and modular congruences.

Concept / Approach:
We work modulo 19. First reduce the bases 15 and 23 modulo 19. Notice that 23 is congruent to 4 modulo 19 and 15 is congruent to −4 modulo 19 because 15 + 4 = 19. That means the two bases are negatives of each other in this modular system. Then, powers of these bases can be related: (−4)²³ = −4²³ when the exponent is odd. Since 23 is an odd exponent, 15²³ ≡ (−4)²³ ≡ −4²³ modulo 19, and 23²³ ≡ 4²³ modulo 19. Adding these two expressions leads to cancellation.

Step-by-Step Solution:
Work modulo 19. Reduce 15 mod 19: 15 ≡ −4 (since 15 + 4 = 19). Reduce 23 mod 19: 23 ≡ 4 (since 23 − 19 = 4). Therefore, 15²³ ≡ (−4)²³ and 23²³ ≡ 4²³ (mod 19). Because 23 is odd, (−4)²³ = −4²³. So 15²³ ≡ −4²³ and 23²³ ≡ 4²³ (mod 19). Now add them: 15²³ + 23²³ ≡ (−4²³) + 4²³ ≡ 0 (mod 19). Hence the remainder when 15²³ + 23²³ is divided by 19 is 0.

Verification / Alternative check:
We can verify the reasoning by checking with smaller exponents. For example, consider 15¹ + 23¹ modulo 19. We get 15 + 23 = 38, and 38 / 19 gives remainder 0. With exponent 3 we have 15³ + 23³. Modulo 19, 15 ≡ −4 and 23 ≡ 4, so (−4)³ + 4³ = −64 + 64 = 0, again giving remainder 0. This consistent pattern for odd exponents supports the same conclusion for exponent 23, since the modular reasoning depends only on parity of the exponent, not its exact size.

Why Other Options Are Wrong:

• 4, 3 and 1: These values might arise if you make mistakes in reducing the bases or overlook that the exponents are odd, but they do not satisfy the exact modular relationships described.
• 7: This is just another incorrect remainder with no basis in the given congruences.

Common Pitfalls:
A common error is attempting to compute large powers directly or using a calculator, which is impractical and unnecessary. Another mistake is forgetting that (−a) raised to an odd power is negative, while raised to an even power is positive. In this problem, misidentifying the parity of 23 leads to wrong signs and incorrect cancellation. Always reduce bases modulo the divisor early, look for symmetry or opposite numbers, and carefully apply power rules for negative bases.

Final Answer:
The remainder when 15²³ + 23²³ is divided by 19 is 0.