As you have observed $$3^4 \equiv 1 \pmod 5$$ This means that $$3^{4k} \equiv 1 \pmod 5$$ Hence, $$3^{80} \equiv 1 \pmod 5 \implies 3^{82} \equiv 3^2 \pmod{5} = 4 \pmod 5$$
In general, $$3^{4k + r} \equiv 3^r \pmod{5}$$
Let's find the LCM of $24$, $15$, and $36$.
$24 = (2^3)(3^1)(5^0)$
$15 = (2^0)(3^1)(5^1)$
$36 = (2^2)(3^2)(5^0)$
The LCM must have all of these factors.
The LCM is $(2^3)(3^2)(5^1)=360$
So now, we need the biggest six digit multiple of $360$ which is $999720$.
You have done this much.
The reason why this works, is that in order for a number to be divisible by all $3$ of these numbers at the same time, it must have the necessary prime factors.
It must have at least $3$ factors of $2$, to be divisible by $24$.
It must have at least $2$ factors of $3$ , to be divisible by $36$.
It must have at least $1$ factor of $5$, to be divisible by $15$.
Note, that if a number satisfies all $3$ of these constraints, it automatically fits all $6$ other constraints. ($3$ actually because all numbers have $p^0$ as a factor)
Notice, when doing these types of calculations, one can be hasty and approximate the solution, and to do so -- multiply all three numbers together. However, for the purposes of accuracy and flexibility, finding the LCM is highly recommended.
Best Answer
Hint: I'd use the Chinese remainder theorem, which says that the mapping $${\Bbb Z}_{45} \rightarrow {\Bbb Z}_5\times {\Bbb Z}_9: x\mapsto (x\mod 5, x\mod 9)$$ is a ring isomorphism.
Thus you can separately divide the large number by 5 and by 9. These remainders can then be combined to obtained the remainder modulo 45.
Modulo 5 the situation is simple, just look at the last digit of the number.
Modulo 9, observe that $10 \equiv 1 \mod 9$ and so $10^n\equiv 1 \mod 9$ for each $n\geq 1$. So modulo 9 you just look for the ''Quersumme'' (cross sum).
So you have to find a number $x$ between 0 and 44 such that $x$ is congruent 4 modulo 5 and congruent $Q$ modulo 9, where $Q$ is the cross sum of your number.
This number is congruent 4 mod 5 and so is one of the following: 4, 9, 14, 19, 24, 29, 34, 39, 44.