Subjects algebra, number theory

Multiple Questions 91C96E

Step-by-step solutions with LaTeX - clean, fast, and student-friendly.

Use the AI math solver

1. **Problem 1:** Given that ก is the smallest natural number and from the image ก + ข + ก is asked. Since ก is the smallest natural number, ก = 1. We are given ข has a remainder 2 (เศษ 2) and ถ has remainder 2 (เศษ 2) but no exact values, so assuming ข = 2 and ก = 1. Calculate ก + ข + ก = 1 + 2 + 1 = 4. 2. **Problem 2:** Find how many natural numbers from 1 to 100 do not contain the digit "9". Step 1: Count numbers from 1 to 100 without digit 9. Step 2: Consider two-digit numbers from 10 to 99. - For the tens digit: can be 1 to 8 (8 options, excluding 9) - For the units digit: can be 0 to 8 (9 options, excluding 9) Step 3: Total two-digit numbers without 9 = $8 \times 9 = 72$. Step 4: For one-digit numbers 1 to 8, all are allowed (8 numbers). Step 5: Number 100 does not contain 9. Step 6: Total numbers without digit 9 = $8 + 72 + 1 = 81$. 3. **Problem 3:** Given the operation \( \triangle \) defined by: $$4 \triangle 2 = 4$$ $$8 \triangle 5 = 9$$ $$10 \triangle 4 = 36$$ $$19 \triangle 13 = 36$$ $$25 \triangle 18 = 49$$ Find $$16 \triangle 4$$. Step 1: Look for a pattern. Step 2: Check if \(a \triangle b = (a - b)^2\). - For 4 △ 2: $(4-2)^2 = 2^2 = 4$ correct. - For 8 △ 5: $(8-5)^2 = 3^2 = 9$ correct. - For 10 △ 4: $(10-4)^2 = 6^2 = 36$ correct. - For 19 △ 13: $(19-13)^2 = 6^2 = 36$ correct. - For 25 △ 18: $(25-18)^2 = 7^2 = 49$ correct. Step 3: Apply to 16 △ 4: $$(16-4)^2 = 12^2 = 144$$ **Final answers:** 1) $4$ 2) $81$ 3) $144$