To solve the problem step by step, we need to find the number of ways to select two numbers \( a \) and \( b \) such that:
1. \( a \) is from the set \( \{2, 4, 6, \ldots, 100\} \)
2. \( b \) is from the set \( \{1, 3, 5, \ldots, 99\} \)
3. The condition \( (a + b) \mod 23 = 2 \) holds.
### Step 1: Identify the sets
The set of even numbers \( a \) can be expressed as:
\[
a = 2n \quad \text{for } n = 1, 2, \ldots, 50
\]
This gives us 50 possible values for \( a \).
The set of odd numbers \( b \) can be expressed as:
\[
b = 2m - 1 \quad \text{for } m = 1, 2, \ldots, 50
\]
This gives us 50 possible values for \( b \).
### Step 2: Set up the equation
We need to find pairs \( (a, b) \) such that:
\[
a + b \equiv 2 \mod 23
\]
This can be rewritten as:
\[
a + b = 23k + 2 \quad \text{for some integer } k
\]
### Step 3: Determine possible values for \( a + b \)
The minimum value of \( a + b \) occurs when \( a = 2 \) and \( b = 1 \):
\[
a + b = 2 + 1 = 3
\]
The maximum value of \( a + b \) occurs when \( a = 100 \) and \( b = 99 \):
\[
a + b = 100 + 99 = 199
\]
### Step 4: Find valid values of \( a + b \)
We need to find all integers \( a + b \) in the range [3, 199] that satisfy \( a + b = 23k + 2 \).
Calculating the values of \( k \):
- For \( k = 0 \): \( a + b = 2 \) (not valid)
- For \( k = 1 \): \( a + b = 25 \)
- For \( k = 2 \): \( a + b = 48 \)
- For \( k = 3 \): \( a + b = 71 \)
- For \( k = 4 \): \( a + b = 94 \)
- For \( k = 5 \): \( a + b = 117 \)
- For \( k = 6 \): \( a + b = 140 \)
- For \( k = 7 \): \( a + b = 163 \)
- For \( k = 8 \): \( a + b = 186 \)
- For \( k = 9 \): \( a + b = 209 \) (not valid)
Thus, the valid sums are \( 25, 48, 71, 94, 117, 140, 163, 186 \).
### Step 5: Count combinations for each valid sum
We need to find the number of pairs \( (a, b) \) for each valid sum.
1. **For \( a + b = 25 \)**:
- Possible pairs: \( (2, 23), (4, 21), (6, 19), (8, 17), (10, 15), (12, 13) \) → 6 ways.
2. **For \( a + b = 48 \)**:
- Possible pairs: \( (2, 46), (4, 44), (6, 42), (8, 40), (10, 38), (12, 36), (14, 34), (16, 32), (18, 30), (20, 28), (22, 26), (24, 24) \) → 12 ways.
3. **For \( a + b = 71 \)**:
- Possible pairs: \( (2, 69), (4, 67), (6, 65), (8, 63), (10, 61), (12, 59), (14, 57), (16, 55), (18, 53), (20, 51), (22, 49), (24, 47), (26, 45), (28, 43), (30, 41), (32, 39), (34, 37), (36, 35), (38, 33), (40, 31), (42, 29), (44, 27), (46, 25), (48, 23), (50, 21), (52, 19), (54, 17), (56, 15), (58, 13), (60, 11), (62, 9), (64, 7), (66, 5), (68, 3), (70, 1) \) → 35 ways.
4. **For \( a + b = 94 \)**:
- Possible pairs: \( (2, 92), (4, 90), (6, 88), (8, 86), (10, 84), (12, 82), (14, 80), (16, 78), (18, 76), (20, 74), (22, 72), (24, 70), (26, 68), (28, 66), (30, 64), (32, 62), (34, 60), (36, 58), (38, 56), (40, 54), (42, 52), (44, 50), (46, 48) \) → 23 ways.
5. **For \( a + b = 117 \)**:
- Possible pairs: \( (2, 115), (4, 113), (6, 111), (8, 109), (10, 107), (12, 105), (14, 103), (16, 101), (18, 99), (20, 97), (22, 95), (24, 93), (26, 91), (28, 89), (30, 87), (32, 85), (34, 83), (36, 81), (38, 79), (40, 77), (42, 75), (44, 73), (46, 71), (48, 69), (50, 67), (52, 65), (54, 63), (56, 61), (58, 59), (60, 57), (62, 55), (64, 53), (66, 51), (68, 49), (70, 47), (72, 45), (74, 43), (76, 41), (78, 39), (80, 37), (82, 35), (84, 33), (86, 31), (88, 29), (90, 27), (92, 25), (94, 23), (96, 21), (98, 19), (100, 17) \) → 42 ways.
6. **For \( a + b = 140 \)**:
- Possible pairs: \( (2, 138), (4, 136), (6, 134), (8, 132), (10, 130), (12, 128), (14, 126), (16, 124), (18, 122), (20, 120), (22, 118), (24, 116), (26, 114), (28, 112), (30, 110), (32, 108), (34, 106), (36, 104), (38, 102), (40, 100), (42, 98), (44, 96), (46, 94), (48, 92), (50, 90), (52, 88), (54, 86), (56, 84), (58, 82), (60, 80), (62, 78), (64, 76), (66, 74), (68, 72), (70, 70), (72, 68), (74, 66), (76, 64), (78, 62), (80, 60), (82, 58), (84, 56), (86, 54), (88, 52), (90, 50), (92, 48), (94, 46), (96, 44), (98, 42), (100, 40) \) → 19 ways.
7. **For \( a + b = 163 \)**:
- Possible pairs: \( (2, 161), (4, 159), (6, 157), (8, 155), (10, 153), (12, 151), (14, 149), (16, 147), (18, 145), (20, 143), (22, 141), (24, 139), (26, 137), (28, 135), (30, 133), (32, 131), (34, 129), (36, 127), (38, 125), (40, 123), (42, 121), (44, 119), (46, 117), (48, 115), (50, 113), (52, 111), (54, 109), (56, 107), (58, 105), (60, 103), (62, 101), (64, 99), (66, 97), (68, 95), (70, 93), (72, 91), (74, 89), (76, 87), (78, 85), (80, 83), (82, 81), (84, 79), (86, 77), (88, 75), (90, 73), (92, 71), (94, 69), (96, 67), (98, 65), (100, 63) \) → 12 ways.
8. **For \( a + b = 186 \)**:
- Possible pairs: \( (2, 184), (4, 182), (6, 180), (8, 178), (10, 176), (12, 174), (14, 172), (16, 170), (18, 168), (20, 166), (22, 164), (24, 162), (26, 160), (28, 158), (30, 156), (32, 154), (34, 152), (36, 150), (38, 148), (40, 146), (42, 144), (44, 142), (46, 140), (48, 138), (50, 136), (52, 134), (54, 132), (56, 130), (58, 128), (60, 126), (62, 124), (64, 122), (66, 120), (68, 118), (70, 116), (72, 114), (74, 112), (76, 110), (78, 108), (80, 106), (82, 104), (84, 102), (86, 100), (88, 98), (90, 96), (92, 94), (94, 92), (96, 90), (98, 88), (100, 86) \) → 6 ways.
### Step 6: Sum the combinations
Now, we add the number of ways for each valid sum:
\[
6 + 12 + 35 + 23 + 42 + 19 + 12 + 6 = 155
\]
### Final Answer
The total number of ways to select the numbers \( a \) and \( b \) such that \( a + b \equiv 2 \mod 23 \) is **155**.
