What will be the remainder when (1234567890123456789)²⁴ is divided by 6561?

Concept: Whether a prime power pm divides a number N depends only on the p-adic valuation vp(N) — the exact power of p dividing N. Two rules govern it: valuations ADD across a product, vp(a·b) = vp(a) + vp(b); and valuations SCALE under a power, vp(Nn) = n·vp(N). Nn is exactly divisible by pm (remainder 0) precisely when vp(Nn) ≥ m; otherwise a genuine nonzero remainder needs separate computation.

Applying it here: 6561 = 38, so we need v3 of the base and compare it to 8 once raised to the 24th power. Factor the base:

1234567890123456789 = 123456789 × (1010 + 1).

1. Digit sum of 123456789 is 45, divisible by 9, so 123456789 = 9 × 13717421. The digit sum of 13717421 is 26, not divisible by 3, so v3(123456789) = 2.

2. 10 ≡ 1 (mod 3), so 1010 + 1 ≡ 110 + 1 ≡ 2 (mod 3) — not divisible by 3 at all, so v3(1010 + 1) = 0.

3. Valuations add across the product: v3(1234567890123456789) = v3(123456789) + v3(1010 + 1) = 2 + 0 = 2.

4. Raising to the 24th power scales the valuation: v3(N24) = 24 × 2 = 48.

Cross-check: Since 48 ≥ 8, N24 is divisible by 38 = 6561, so the remainder is 0. Independently, computing the full number modulo 27 directly gives 1234567890123456789 mod 27 = 9 — divisible by 9 but not by 27 — which confirms v3(N) = 2 exactly as found by factoring, without relying on that factorisation.

Result: The remainder when (1234567890123456789)24 is divided by 6561 is 0.