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Gia sư môn Toán » Học toán lớp 6 trên mạng, Toán Lớp 7

Các bài toán vận dụng nguyên lý Di-rich-le – Bài 1

[Nguyên lý Di-rich-le trong toán THCS] – Đề bài: Cho dãy số: 10, 102, 103, … 1020. Chứng minh rằng tồn tại một số chia cho 19 dư 1.

Hướng dẫn giải:

Theo nguyên lý Di-rich-lê tồn tại 2 số trong 20 số trên khi chia cho 19 có cùng số dư.

Giả sử có hai số: 10n, 10m (n < m) khi chia cho 19 có cùng số dư. Suy ra hiệu của hai số chia hết cho 19.

  • 10m – 10n ⋮ 19
  • 10n.(10m-n – 1) ⋮ 19, mà 10n không chia hết cho 19
  • 10m-n – 1 ⋮ 19
  • 10m-n – 1 = 19k (k ∈ N)
  • 10m-n = 19k + 1
  • Tồn tại một số trong dãy số trên chia cho 19 dư 1 (đpcm).

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