2017寒假猿辅导初等数论-2: "带余除法"作业题解答

 

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1. 计算: $(5767, 4453)$, $(3141, 1592)$, $(136, 221, 391)$.

解答: $$(5767, 4453) = (4453, 1314) = (1314, 511) = (511, 292) = (292, 219) = (219, 73) = 73.$$ $$(3141, 1592) = (1592, 1549) = (1549, 43) = (43, 1) = 1.$$ $$(136, 221, 391) = (136, (221, 391)) = (136, (221, 170)) = (136, (170, 51)) = (136, 17) = 17.$$

 

 

2. 设 $n$ 是整数, 证明: $$(12n + 5, 9n + 4) = 1.$$ 解答:

由Bezout恒等式, $$4(9n+4) - 3(12n+5) = 1\Rightarrow (12n + 5, 9n + 4) = 1.$$

 

 

3. 设 $m, n\in\mathbf{N^*}$, 且 $m$ 是奇数, 证明: $$\left(2^m - 1, 2^n + 1\right) = 1.$$ 解答:

设 $d = \left(2^m - 1, 2^n + 1\right)\Rightarrow 2^m - 1 = da$, $2^n + 1 = db$, $(a, b) = 1$. $$\Rightarrow 2^{mn} = (da + 1)^n = (db - 1)^m$$ $$\Rightarrow dA + 1 = dB - 1\Rightarrow d(B-A) = 2 \Rightarrow d\ |\ 2 \Rightarrow d = 1, 2.$$ 易知 $d$ 是奇数, 因此 $d = 1$.

 

 

4. 设 $(a, b) = 1$, 证明: $$\left(a^2 + b^2, ab\right) = 1.$$ 解答: $$(a, b) = 1\Rightarrow (a^2, b) = 1 \Rightarrow (a^2 + b^2, b) = 1.$$ 同理可得 $(a^2 + b^2, a) = 1$. 因此 $(a^2 + b^2, ab) = 1$.

 

 

5. 证明: 若 $a, b\in\mathbf{N^*}$, 那么数列: $a$, $2a$, $\cdots$, $ba$ 中能被 $b$ 整除的项的个数等于 $(a, b)$.

解答:

令 $(a, b) = d$, $a = da_1$, $b = db_1$, $(a_1, b_1) = 1$. 则原数列为 $$da_1, da_2, \cdots, db_1da_1,$$ 同时除以 $b$ 得 $$\frac{a_1}{b_1}, \frac{2a_1}{b_1}, \cdots, \frac{(db_1)a_1}{b_1},$$ 其中为整数的项为 $\dfrac{(ib_1)a_1}{b_1}$, 其中 $i = 1, 2, \cdots, d$ 总计 $d$ 个.