Animated Solution for Mathematics - Binomial Theorem: $3\times 7^{22}+2\times 10^{22}-44$ when divided by 18 leaves the remainder.

To find the remainder of the expression $E=3\times 7^{22}+2\times 10^{22}-44$ when divided by $18$, we avoid brute force calculation. Instead, we utilize the Binomial Theorem to simplify the bases relative to the modulus $18$.

We observe that $7=1+6$ and $10=1+9$. Since both $6$ and $9$ are related to $18$, this transformation allows us to isolate the significant terms in the expansion.

We express the first part as $3(1+6{)}^{22}$. Applying the Binomial Theorem:

Note that $6^2=36$, which is a multiple of $18$. Consequently, all terms containing $6^k$ for $k\ge 2$ are congruent to $0(\mathrm{mod}18)$.

This simplifies the expression to:

Dividing $399$ by $18$, we find $399=18\times 22+3$. Thus, $3\times 7^{22}\equiv 3(\mathrm{mod}18)$.

Next, we consider $2\times 10^{22}=2(1+9{)}^{22}$. Expanding this, we get:

Since $9^2=81$, and $2\times 81=162=18\times 9$, all terms from the quadratic term onwards are congruent to $0(\mathrm{mod}18)$.

We are left with:

Dividing $398$ by $18$, we find $398=18\times 22+2$. Thus, $2\times 10^{22}\equiv 2(\mathrm{mod}18)$.

We now combine our results to evaluate the full expression $E(\mathrm{mod}18)$:

To obtain a positive remainder, we add a sufficient multiple of $18$ to $-39$. Adding $54$ (which is $18\times 3$):

The final remainder is $15$.