This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To solve the congruence d × d + 2 = 14 8: Step 1: Rewrite the equation. The equation can be written as: d^2 + 2 14 8 Step 2: Subtract 2 from both sides of the congruence. d^2 14 - 2 8 d^2 12 8 Step 3: Reduce the right side modulo 8. To find 12 8, we divide 12 by 8 and find the remainder: 12 = 1 × 8 + 4. So, 12 4 8. The congruence becomes: d^2 4 8 Step 4: Find values of d that satisfy the congruence. We need to find integers d such that d^2 leaves a remainder of 4 when divided by 8. We can test values for d from 0 to 7: • If d=0, d^2 = 0^2 = 0. 0 4 8. • If d=1, d^2 = 1^2 = 1. 1 4 8. • If d=2, d^2 = 2^2 = 4. 4 4 8. This is a solution. • If d=3, d^2 = 3^2 = 9. 9 1 8. 1 4 8. • If d=4, d^2 = 4^2 = 16. 16 0 8. 0 4 8. • If d=5, d^2 = 5^2 = 25. 25 1 8. 1 4 8. • If d=6, d^2 = 6^2 = 36. 36 4 8. This is a solution. • If d=7, d^2 = 7^2 = 49. 49 1 8. 1 4 8. The values of d that satisfy the congruence are d 2 8 and d 6 8. The final answer is d 2 8 or d 6 ±od8.