If the unit digit in the product of [tex][tex]$163 \times 87 \times \square \times 239$[/tex][/tex] is 1, then the unit digit that should replace [tex]$\square$[/tex] will be:

A. 9
B. 3



Answer :

Let's solve the problem step-by-step.

We need the final digit of the product [tex]\(163 \times 87 \times \square \times 239\)[/tex] to be 1. Let's first examine the unit digit of the product [tex]\(163 \times 87 \times 239\)[/tex].

1. Determine the Last Digit of the Product [tex]\(163 \times 87 \times 239\)[/tex]:
- The unit digit of 163 is 3.
- The unit digit of 87 is 7.
- The unit digit of 239 is 9.

To find the unit digit of their product, we multiply their unit digits:

[tex]\[3 \times 7 = 21 \quad \text{(unit digit is 1)}\][/tex]

[tex]\[1 \times 9 = 9 \quad \text{(unit digit is 9)}\][/tex]

So, the unit digit of the product [tex]\(163 \times 87 \times 239\)[/tex] is 9.

2. Find the Unit Digit for [tex]\(\square\)[/tex] to Make the Final Product's Unit Digit 1:
- Let [tex]\(d\)[/tex] be the digit in place of [tex]\(\square\)[/tex].
- We need to find a digit [tex]\(d\)[/tex] such that when 9 (the unit digit of the intermediate product) multiplied by [tex]\(d\)[/tex], the unit digit of the resulting product is 1.

Let's check the products of 9 with digits 0 through 9 until we get a unit digit of 1:

[tex]\[ \begin{align*} 9 \times 0 & = 0 \quad (\text{unit digit is 0}) \\ 9 \times 1 & = 9 \quad (\text{unit digit is 9}) \\ 9 \times 2 & = 18 \quad (\text{unit digit is 8}) \\ 9 \times 3 & = 27 \quad (\text{unit digit is 7}) \\ 9 \times 4 & = 36 \quad (\text{unit digit is 6}) \\ 9 \times 5 & = 45 \quad (\text{unit digit is 5}) \\ 9 \times 6 & = 54 \quad (\text{unit digit is 4}) \\ 9 \times 7 & = 63 \quad (\text{unit digit is 3}) \\ 9 \times 8 & = 72 \quad (\text{unit digit is 2}) \\ 9 \times 9 & = 81 \quad (\text{unit digit is 1}) \end{align*} \][/tex]

Therefore, the digit [tex]\(d\)[/tex] that makes the unit digit of the product 1 is 9.

Conclusion: The digit that should be in place of [tex]\(\square\)[/tex] to make the last digit of the product [tex]\(163 \times 87 \times \square \times 239\)[/tex] be 1 is 9.

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