Đáp án + Giải thích các bước giải:

 The number will be as follows: $74747474\underbrace{...}_{168\text{ "74"s}}7$

So the product of digits is like this: $7 \times 4 \times 7 \times 4 \times 7 \times 4 \times 7 \times 4 \times \underbrace{...}_{168\text{"7}\times\text{4"}s} \times 7$ or just $28 \times 28 \times 28 \times 28 \times \underbrace{...}_{168\text{ "28"}s} \times 7$

Now we split the product into two parts: the first part is the part with only the multiplication of $28$s and the second part is the lone number $7$.

In the first part, we can find out that there are $172$ number $28$s. The below table shows the unit digit of the products when $28$ is multiplied by itself $n$ times, or simply the result of $28^n$:

\begin{array}{|c|c|c|}\hline \text{n}&\text{1}&\text{2}&\text{3}&\text{4}&\text{5}&\text{6}&\text{7}&\text{8}&\text{9}&\text{...}\\\hline \text{unit digit of product}&\text{8}&\text{4}&\text{2}&\text{6}&\text{8}&\text{4}&\text{2}&\text{6}&\text{8}&\text{...}\\\hline \end{array}

You may notice that the every $4$ number, the end digits become $8,4,2,6$ again. This gives us a pattern: 

- If $\dfrac{n}{4}$ has a remainder of $1$, the unit digit of the product is $8.$

- If $\dfrac{n}{4}$ has a remainder of $2$, the unit digit of the product is $4.$

- If $\dfrac{n}{4}$ has a remainder of $3$, the unit digit of the product is $2.$

- If $\dfrac{n}{4}$ has no remainder, the unit digit of the product is $6.$

We now consider the product of $28^{172}$. $172$ divided by $4$ gives us $43$ with no remainder, so the unit digit is $6$. Now the unit digit product of all $345$ digits in the original number is just the unit digit of the first part multiplied by the second part, or $6 \times 7$. $6 \times 7$ is $42$, so the unit digit of the product of digits in the original number is $2$.

If the number has `345` but the number is of the form ''74747474...'' then the number `7` is odd digits
Ex: `345;343;341;.....1` are the digits `7`
then `14` are even digits
Ex: `344;340;398;.....2` are the digits `4`
⇒ Units row is the first digit (number `1`)
⇒ The unit line of this product is 7

Dịch :

Nếu số có `354` mà số có dạng `'' 74747474 ... ''` thì số 7 là các chữ số lẻ
Ví dụ`: 345; 343; 341; ..... 1` là các chữ số `7`
thì 4 là chữ số chẵn
Ví dụ`: 344; 340; 398; ..... 2` là các chữ số `4` ⇒ Hàng đơn vị là chữ số đầu tiên (số `1`)
⇒ Dòng đơn vị của sản phẩm này là `7`

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