The number of distinct integer solutions (x, y) of the equation |x + y| + |x – y| = 2, is
Solution
Since two absolute values are added, for the equation to be true we can have three combinations of \(|x + y|, |x – y|\): \[ \{2, 0\}, \{0, 2\}, \{1, 1\} \]
Cases and Corresponding Solutions:
| Case | Conditions | Solution(s) |
|---|---|---|
| Case 1 | \( x + y = 2 \) and \( x – y = 0 \) | \( \{x, y\} = \{1, 1\} \) |
| Case 2 | \( x + y = -2 \) and \( x – y = 0 \) | \( \{x, y\} = \{-1, -1\} \) |
| Case 3 | \( x + y = 0 \) and \( x – y = 2 \) | \( \{x, y\} = \{1, -1\} \) |
| Case 4 | \( x + y = 0 \) and \( x – y = -2 \) | \( \{x, y\} = \{-1, 1\} \) |
| Case 5 | \( x + y = 1 \) and \( x – y = 1 \) | \( \{x, y\} = \{1, 0\} \) |
| Case 6 | \( x + y = 1 \) and \( x – y = -1 \) | \( \{x, y\} = \{0, 1\} \) |
| Case 7 | \( x + y = -1 \) and \( x – y = 1 \) | \( \{x, y\} = \{0, -1\} \) |
| Case 8 | \( x + y = -1 \) and \( x – y = -1 \) | \( \{x, y\} = \{-1, 0\} \) |
✅ Hence, there are 8 distinct integer solutions for \(\{x, y\}\).
If 10^68 is divided by 13, the remainder is
1) 8 2) 9 3) 4 4) 5
Solution
Let’s solve 10^68 ÷ 13 — find the remainder using the cycle method:
Step 1: Find powers of 10 modulo 13
- 10¹ mod 13 = 10
- 10² mod 13 = (10 × 10) mod 13 = 100 mod 13
100 ÷ 13 = 7 remainder 9 → 10² mod 13 = 9 - 10³ mod 13 = (10² × 10) mod 13 = (9 × 10) mod 13 = 90 mod 13
90 ÷ 13 = 6 remainder 12 → 10³ mod 13 = 12 - 10⁴ mod 13 = (12 × 10) mod 13 = 120 mod 13
120 ÷ 13 = 9 remainder 3 → 10⁴ mod 13 = 3 - 10⁵ mod 13 = (3 × 10) mod 13 = 30 mod 13
30 ÷ 13 = 2 remainder 4 → 10⁵ mod 13 = 4 - 10⁶ mod 13 = (4 × 10) mod 13 = 40 mod 13
40 ÷ 13 = 3 remainder 1 → 10⁶ mod 13 = 1
✅ The cycle length is 6:
10^6 mod 13 = 1
Step 2: Find position in cycle
10^68 mod 13 = 10^(68 mod 6) mod 13
68 ÷ 6 = 11 remainder 2
So 10^68 mod 13 = 10² mod 13 = 9
✅ Final answer:
👉 The remainder is 9
Correct option: 2) 9
Find the number of all positive integers up to 500 with non-repeating digits.
| Type of Number | Digit Positions | Choices at Each Step | Reasoning | Count |
|---|---|---|---|---|
| 1-digit | _ | 1-9 → 9 choices | Any digit 1-9 is valid | 9 |
| 2-digit | Tens digit | 1-9 → 9 choices | Tens digit can’t be 0 | 81 |
| Units digit | 0-9 except tens digit → 9 choices | Can’t repeat tens digit | ||
| 3-digit (100-499) | Hundreds digit | 1-4 → 4 choices | First digit ≤ 4 so number ≤ 499 | 288 |
| Tens digit | 0-9 except hundreds digit → 9 choices | Must differ from hundreds digit | ||
| Units digit | 0-9 except hundreds & tens digit → 8 choices | Must differ from both |
Final Total: 9 (1-digit) + 81 (2-digit) + 288 (3-digit) = 378
