1. **Problem:** Show that the square of any integer can be written in the form $4q$ or $4q+1$. 2. **Formula and rules:** The division algorithm states that for any integer $a$ and positive integer $b$, there exist unique integers $q$ and $r$ such that $$a = bq + r$$ where $0 \leq r < b$. 3. **Step 1: Express any integer $n$ in terms of division by 4:** By the division algorithm, for any integer $n$, there exist integers $q$ and $r$ such that $$n = 4q + r$$ where $r \in \{0,1,2,3\}$. 4. **Step 2: Square $n$:** $$n^2 = (4q + r)^2 = 16q^2 + 8qr + r^2$$ 5. **Step 3: Consider $r^2$ modulo 4:** - If $r=0$, $r^2=0$ and $n^2 = 16q^2 + 8q\cdot0 + 0 = 4(4q^2 + 0)$ which is of the form $4q$. - If $r=1$, $r^2=1$ and $$n^2 = 16q^2 + 8q + 1 = 4(4q^2 + 2q) + 1$$ which is of the form $4q + 1$. - If $r=2$, $r^2=4$ and $$n^2 = 16q^2 + 16q + 4 = 4(4q^2 + 4q + 1)$$ which is of the form $4q$. - If $r=3$, $r^2=9$ and $$n^2 = 16q^2 + 24q + 9 = 4(4q^2 + 6q + 2) + 1$$ which is of the form $4q + 1$. 6. **Conclusion:** The square of any integer is either $4q$ or $4q + 1$. --- 7. **Problem:** Find $q$ and $r$ for the division of 148 by -17 and -275 by 39 using the division algorithm. 8. **Note:** The divisor $b$ must be positive for the division algorithm. So for 148 divided by -17, we consider dividing by 17 and adjust signs accordingly. 9. **For 148 divided by 17:** Divide 148 by 17: $$17 \times 8 = 136 \quad \Rightarrow \quad 148 - 136 = 12$$ So, $$148 = 17 \times 8 + 12$$ Thus, $q=8$, $r=12$. 10. **For -275 divided by 39:** Divide -275 by 39: Find $q$ such that $$-275 = 39q + r$$ with $0 \leq r < 39$. Try $q = -8$: $$39 \times (-8) = -312$$ $$r = -275 - (-312) = 37$$ which satisfies $0 \leq r < 39$. So, $$-275 = 39 \times (-8) + 37$$ Thus, $q = -8$, $r = 37$. --- "q_count": 2