Divisibility check — Evaluate 19^5 + 21^5 and state whether it is divisible by 10, by 20, by both, or by neither. Use modular reasoning (units and mod 4).

Introduction / Context:
Instead of computing huge powers, divisibility questions can be solved using modular arithmetic. Here you assess 19^5 + 21^5 with respect to 10 and 20. Divisibility by 10 requires a last digit 0 (i.e., divisibility by 2 and 5), while divisibility by 20 requires divisibility by both 4 and 5. We will use units-digit patterns and modular reductions to conclude quickly.

Given Data / Assumptions:

• Expression: 19^5 + 21^5.
• We test divisibility by 10 and by 20.
• We use modular arithmetic (mod 10 for last digit, mod 4 for the factor 4, and mod 5 for the factor 5 if needed).

Concept / Approach:
Use periodicity of last digits for powers to handle mod 10. For mod 4, reduce bases to their remainders modulo 4. Finally, confirm the factor 5 via mod 5 or simple observation from the last digit. If the number is divisible by both 4 and 5, it is divisible by 20.

Step-by-Step Solution:
Units digits: 19 → 9, 21 → 1. For odd exponents, 9^odd ends in 9 and 1^odd ends in 1.Hence last digit of 19^5 + 21^5 is 9 + 1 = 10 → ends with 0 → divisible by 10.Check mod 4: 19 ≡ 3 (mod 4), 3^2 ≡ 1, so 3^5 ≡ 3 (mod 4). Also, 21 ≡ 1 (mod 4), 1^5 ≡ 1. Sum ≡ 3 + 1 = 0 (mod 4).Since the sum is divisible by 4 and ends with 0 (divisible by 5), it is divisible by 20.

Verification / Alternative check:
Observe mod 5: 19 ≡ -1 (mod 5) so 19^5 ≡ -1; 21 ≡ 1 (mod 5) so 21^5 ≡ 1; sum ≡ 0 (mod 5). Combined with the mod 4 result, divisibility by 20 is confirmed via the lcm approach.

Why Other Options Are Wrong:

• Only 10 / Only 20 / Neither: The sum satisfies divisibility by both 10 and 20 as shown, so these are incorrect.
• Only 5: Too weak; the sum is divisible by 5 but also by 4, hence by 20 and 10.

Common Pitfalls:
Trying to compute the full powers; ignoring mod 4; assuming “ends with 0” implies exactly 10 but not checking for 20; overlooking periodicity of units digits.

Final Answer:
Both 10 and 20