To find the fractional part of the expression, we need to find the remainder when $$3^{2026}$$ is divided by 80.
We rewrite the base of the exponent to match a number close to a multiple of 80 by noticing that $$3^4 = 81 = 80 + 1$$.
First, we split the exponent to express it in terms of a power of 4 by writing $$2026 = 4 \times 506 + 2$$.
This allows us to separate the terms into $$3^{2026} = 3^2 \times (3^4)^{506} = 9 \times (81)^{506}$$.
Substituting $$81 = 80 + 1$$ into the expression yields $$9 \times (80 + 1)^{506}$$.
Applying the Binomial Theorem to expand the two-term expression yields $$9 \times \left[ \binom{506}{0}(80)^{506} + \binom{506}{1}(80)^{505} + \dots + \binom{506}{505}(80)^1 + \binom{506}{506}(1)^{506} \right]$$.
Every term inside the brackets except the last one contains a factor of 80, so we can group them together as an integer multiple $$80M$$. This simplifies our expression to $$9 \times (80M + 1) = 720M + 9$$.
Dividing this entire expression by 80 gives $$\frac{720M + 9}{80} = 9M + \frac{9}{80}$$. Since $$9M$$ is an integer, the fractional part of the number is exactly $$\frac{9}{80}$$.
Comparing this to the given form $$\frac{k}{80}$$ tells us that $$k = 9$$. The integer 9 can be factored into prime numbers as $$3^2$$, which means it has only 1 distinct prime factor, namely 3.
Hence, the final answer is 1
91- 6303239042
