I'm working in the following chinese's remainder theorem exercise:
Calculate
$$1434661\cdot 3785648743-10020304\cdot 54300201$$
You are told that the answer is a positive integer less than $90$. Compute
the answer ${\mod 10}$ and ${\mod 9}$, then combine to get the answer.
I've planned a system based on the criterion that the answer is less than $90$ with $d = 1434661*3785648743-10020304*54300201$:
$$\begin{cases}d \equiv 9\pmod{10}\\
d \equiv x \pmod{9}\end{cases}$$
But I'm not sure about the system to be correct, any help will be really appreciated.
Best Answer
Well, you were told what to do so ... do it.
$$1434661\equiv 1 \pmod {10};⋅3785648743\equiv 3\pmod{10};$$ $$ 10020304\equiv 4\pmod{10}; ⋅54300201\equiv 1\pmod {10}.$$
So $$1434661⋅3785648743−10020304⋅54300201 \equiv 1\cdot 3-4\cdot 1\equiv -1\equiv 9 \pmod{10}.$$
And using the sum of digits for multiples of $9$ rule:
$$1434661\equiv1+4+3+4+6+6+1 \equiv 25\equiv 2+5 \equiv 7 \pmod {9};$$ $$3785648743\equiv 3+7+8+5+6+4+8+7+4+3\equiv 55 \equiv 5+5$$ $$ \equiv 10 \equiv 1\pmod{9}; $$ $$10020304\equiv1+0+0+2+0+3+0+4\equiv 10 \equiv 1\pmod{9}; $$ $$54300201\equiv 5+4+3+0+0+2+0+1\equiv 15 \equiv 1+5 \equiv 6\pmod {9}.$$
So $$1434661⋅3785648743−10020304⋅54300201 \equiv 7\cdot 1-1\cdot 6\equiv 1 \pmod{9}.$$
So there is a unique $0\le x < 90$ so that $x \equiv -1 \pmod {10}$ and $x \equiv 1 \pmod {9}$.
$x = 9,19,29, .....,$ or $ 89$ and $x = 1, 10 ,19,28,....., $ or $82,$
so $x = 19$.
So $1434661⋅3785648743−10020304⋅54300201\equiv 19 \pmod {90}.$