Multiplication Is Tedious

Relevant wiki: Decimals

Method 1 : Multiply the 3 decimals out

With enough patience, we can see that $1.2\times3.45\times6.789 = 28.106460$ , by keeping track the total number of decimal places ( $1+2+3 = 6$ ). And we need to find out the minimum value of $n$ such that $28.106460 \times 10^n$ is an integer.

$\begin{aligned} 28.106460 \times10^1 &=& 281.06460 \qquad \text{ not an integer} \\ 28.106460 \times10^2 &=& 2810.6460 \qquad \text{ not an integer} \\ 28.106460 \times10^3 &=& 28106.460 \qquad \text{ not an integer} \\ 28.106460 \times10^4 &=& 281064.60 \qquad \text{ not an integer} \\ 28.106460 \times10^5 &=& 2810646.0 \qquad \text{ is an integer!} \end{aligned}$

Hence, our answer is $\boxed5$ .

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Method 2 : Modular arithmetic

Let the fractional part of $1.2\times3.45\times6.789$ be $0.A_1 A_2 A_3 A_4 A_5 A_6 A_7 \ldots$ , where $A_1, A_2,A_3,\ldots$ are single digits.

Notice that we're multiplying decimal numbers, and the total number of decimal places involved is 6, so all the digits after the sixth decimal places are all 0's, $A_7 = A_8 = A_9 = \cdots = 0$ .

$1.2\times3.45 \times6.789 = \dfrac{12}{10}\times\dfrac{345}{100} \times \dfrac{6789}{1000} = \dfrac1{10^6} ( 12\times 345\times7689 ) \;$

And so, our 6-digit integer, $\overline{A_1 A_2 A_3 A_4 A_5 A_6}$ is the last 6 digits of $12\times 345\times7689$ .

Equivalently, $A_6$ is the last digit of $12\times 345\times6789$ , which can be found by multiplying the last digits: $2\times5\times9 \equiv 5 \pmod{10}$ .

And, the 2-digit number $\overline{A_5 A_6 }= \overline{A_5 0}$ is the last two digits of $12\times 345\times6789$ , which can be found by multiplying the last two digits, $12\times45\times89 \equiv 60 \pmod{100}$ .

Relevant wiki: Decimals

1.2 ends with 2 and 3.45 ends with 5 due to which $1.2 × 3.45$ forms a number with $3-1=2$ decimal places. Now this number formed has 2 decimal places and 6.789 has 3 decimal places. So, their product is a number with 5 decimal places. So, ${10}^5$ has to be multiplied to make this number an integer. So, $n = \boxed{5}$ .