1. **Problem 1:** Find how many integers $M$ between 16 and 27 have an even number of factors. 2. **Step 1:** Recall that the number of factors of a number is even unless the number is a perfect square (which has an odd number of factors). 3. **Step 2:** List integers between 16 and 27: 17, 18, 19, 20, 21, 22, 23, 24, 25, 26. 4. **Step 3:** Identify perfect squares in this range: $16 = 4^2$ (excluded since 16 is not between 16 and 27), $25 = 5^2$. 5. **Step 4:** Since 25 is the only perfect square in the range, it has an odd number of factors. 6. **Step 5:** Total numbers between 16 and 27 (excluding 16 and 27) are 10; subtract 1 perfect square gives 9 numbers with even number of factors. 7. **Problem 2:** Given $ab=18$, $bc=39$, $ac=78$, find the value of $(a^2 - b^2 - c^2 + 3abc)(a - b + c)$. 8. **Step 1:** From the given, find $a$, $b$, and $c$. 9. **Step 2:** Multiply all three equations: $(ab)(bc)(ac) = 18 imes 39 imes 78$. 10. This equals $a^2 b^2 c^2 = (abc)^2$. 11. Calculate $18 imes 39 = 702$, then $702 imes 78 = 54756$. 12. So, $(abc)^2 = 54756$ and $abc = \sqrt{54756}$. 13. **Step 3:** Find $a$, $b$, and $c$ individually. 14. From $ab=18$ and $ac=78$, divide $ac$ by $ab$: $\frac{ac}{ab} = \frac{78}{18} = \frac{13}{3} = \frac{c}{b}$. 15. So, $c = \frac{13}{3} b$. 16. From $bc=39$, substitute $c$: $b \times \frac{13}{3} b = 39 \Rightarrow \frac{13}{3} b^2 = 39$. 17. Multiply both sides by 3: $13 b^2 = 117$. 18. Divide both sides by 13: $b^2 = 9$. 19. So, $b = 3$ (taking positive since natural numbers). 20. Then $c = \frac{13}{3} \times 3 = 13$. 21. From $ab=18$, $a = \frac{18}{b} = \frac{18}{3} = 6$. 22. **Step 4:** Calculate each term: - $a^2 = 6^2 = 36$ - $b^2 = 3^2 = 9$ - $c^2 = 13^2 = 169$ - $abc = 6 \times 3 \times 13 = 234$ 23. Substitute into expression: $$(a^2 - b^2 - c^2 + 3abc)(a - b + c) = (36 - 9 - 169 + 3 \times 234)(6 - 3 + 13)$$ 24. Simplify inside first parentheses: $36 - 9 - 169 = -142$ $3 \times 234 = 702$ Sum: $-142 + 702 = 560$ 25. Simplify second parentheses: $6 - 3 + 13 = 16$ 26. Multiply: $560 \times 16 = 8960$ 27. **Problem 3:** Find the second number from the left on the 15th row in a snake pattern where odd rows increase left to right and even rows decrease left to right. 28. **Step 1:** Each row has 5 numbers. 29. **Step 2:** Total numbers before 15th row: $14 \times 5 = 70$. 30. **Step 3:** The 15th row is odd, so numbers increase left to right starting from $70 + 1 = 71$. 31. **Step 4:** The second number from the left is $71 + 1 = 72$. **Final answers:** 1. 9 2. 8960 3. 72