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Giải bài 9 trang 29 vở thực hành Toán 8

Biết số tự nhiên a chia 3 dư 2. Chứng minh rằng \({a^2}\) chia 3 dư 1.

🌊Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

🧸Biết số tự nhiên a chia 3 dư 2. Chứng minh rằng \({a^2}\) chia 3 dư 1.

Phương pháp giải - Xem chi tiết

ꦇSử dụng hằng đẳng thức bình phương của một tổng: \({\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\)

Lời giải chi tiết

ღVì a chia 3 dư 2 nên ta có thể viết \(a = 3n + 2,n\; \in \;\mathbb{N}\). Ta có

🐲\(\begin{array}{*{20}{l}}{{a^2}\; = {{\left( {3n + 2} \right)}^2}\; = 9{n^2}\; + 12n + 4}\\{ = 9{n^2}\; + 12n + 3 + 1}\\{ = 3.\left( {3{n^2}\; + 4n + 1} \right) + 1.}\end{array}\)

ꦬVì \(3\; \vdots \;3\) nên tích \(3.\left( {3{n^2}\; + 4n + 1} \right)\) chia hết cho 3 và do đó \(3.\left( {3{n^2}\; + 4n + 1} \right) + 1\) chia 3 dư 1. Vậy \({a^2}\) chia 3 dư 1.

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