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Giải bài 7 trang 18 vở thực hành Toán 8

Chứng minh rằng nếu m và n nhận các giá trị nguyên tùy ý thì biểu thức

ꦛTổng 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

Chứng minh rằng nếu m và n nhận các giá trị nguyên tùy ý thì biểu thức

🦩\(K = \left( {5m + 1} \right)\left( {5n-2} \right) + \left( {5m-2} \right)\left( {5n + 1} \right) + 4\)

luôn có giá trị là số nguyên chia hết cho 5.

Phương pháp giải - Xem chi tiết

Sử dụng quy tắc nhân hai đa thức: Muốn nhân một đa thức với một đa thức, ta nhân mỗi hạng tử của đa thức này với từng hạng tử của đa thức kia rồi cộng các tích với nhau.

Lời giải chi tiết

Ta biến đổi biểu thức K như sau:

🦹\(\begin{array}{*{20}{l}}{K = \left( {5m + 1} \right)\left( {5n-2} \right) + \left( {5m-2} \right)\left( {5n + 1} \right) + 4}\\{ = \left( {25mn-10m + 5n-2} \right) + \left( {25mn + 5m-10n-2} \right) + 4}\\{ = 50mn-5m-5n = 5\left( {10mn-m-n} \right).}\end{array}\)

Từ kết quả trên, ta thấy K có dạng K = 5k, trong đó k = 10mn – m – n.Ta thấy K luôn có giá trị là số nguyên tại mọi giá trị nguyên của m và n.Do đó K luôn có giá trị là số nguyên chia hết cho 5.

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