1. **Problem:** Determine between which two integers the number $\sqrt{3}$ lies. Step 1: Identify perfect squares around 3. The perfect squares near 3 are $1 = 1^2$ and $4 = 2^2$. Step 2: Since $1 < 3 < 4$, it follows that $1 < \sqrt{3} < 2$. 2. **Problem:** Determine between which two integers the number $\sqrt{110}$ lies. Step 1: Identify perfect squares around 110. The perfect squares near 110 are $100 = 10^2$ and $121 = 11^2$. Step 2: Since $100 < 110 < 121$, it follows that $10 < \sqrt{110} < 11$. 3. **Problem:** Determine between which two integers the number $\sqrt{5}$ lies. Step 1: Identify perfect squares around 5. The perfect squares near 5 are $4 = 2^2$ and $9 = 3^2$. Step 2: Since $4 < 5 < 9$, it follows that $2 < \sqrt{5} < 3$. 4. **Problem:** Determine between which two integers the number $\sqrt{36}$ lies. Step 1: Recognize that $36 = 6^2$. Step 2: Therefore, $\sqrt{36} = 6$ exactly. 5. **Problem:** Determine between which two integers the number $\sqrt{12}$ lies. Step 1: Identify perfect squares around 12. The perfect squares near 12 are $9 = 3^2$ and $16 = 4^2$. Step 2: Since $9 < 12 < 16$, it follows that $3 < \sqrt{12} < 4$. 6. **Problem:** Determine between which two integers the number $\sqrt{50}$ lies. Step 1: Identify perfect squares around 50. The perfect squares near 50 are $49 = 7^2$ and $64 = 8^2$. Step 2: Since $49 < 50 < 64$, it follows that $7 < \sqrt{50} < 8$. 7. **Problem:** Determine between which two integers the number $\sqrt{22}$ lies. Step 1: Identify perfect squares around 22. The perfect squares near 22 are $16 = 4^2$ and $25 = 5^2$. Step 2: Since $16 < 22 < 25$, it follows that $4 < \sqrt{22} < 5$. 8. **Problem:** Determine between which two integers the number $\sqrt{80}$ lies. Step 1: Identify perfect squares around 80. The perfect squares near 80 are $64 = 8^2$ and $81 = 9^2$. Step 2: Since $64 < 80 < 81$, it follows that $8 < \sqrt{80} < 9$. 9. **Problem:** Determine between which two integers the number $\sqrt{35}$ lies. Step 1: Identify perfect squares around 35. The perfect squares near 35 are $25 = 5^2$ and $36 = 6^2$. Step 2: Since $25 < 35 < 36$, it follows that $5 < \sqrt{35} < 6$. 10. **Problem:** Determine between which two integers the number $\sqrt{99}$ lies. Step 1: Identify perfect squares around 99. The perfect squares near 99 are $81 = 9^2$ and $100 = 10^2$. Step 2: Since $81 < 99 < 100$, it follows that $9 < \sqrt{99} < 10$. **Final answers:** 1. $1 < \sqrt{3} < 2$ 2. $10 < \sqrt{110} < 11$ 3. $2 < \sqrt{5} < 3$ 4. $\sqrt{36} = 6$ 5. $3 < \sqrt{12} < 4$ 6. $7 < \sqrt{50} < 8$ 7. $4 < \sqrt{22} < 5$ 8. $8 < \sqrt{80} < 9$ 9. $5 < \sqrt{35} < 6$ 10. $9 < \sqrt{99} < 10$