"I hold some pebbles, not too many, When I group them in 3's, one stays with me. Try pairing them up — it simply won't do, A stubborn odd pebble remains in my view. Group them by 5, yet one's still around, But grouping by seven, perfection is found. More than one hundred would be far too bold, Can you tell me the number of pebbles I hold?"
Given Conditions:
- Remainder 1 when divided by 3
- Odd number (cannot be paired)
- Remainder 1 when divided by 5
- Divisible by 7
- Less than 100
Solution:
Step 1: Find numbers < 100 divisible by 7 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98
Step 2: Filter odd numbers 7, 21, 35, 49, 63, 77, 91
Step 3: Check remainder 1 when divided by 3 7 (7÷3=2r1), 49 (49÷3=16r1), 91 (91÷3=30r1)
Step 4: Check remainder 1 when divided by 5 Only 91 (91÷5=18r1)
Answer: 91 pebbles
Explanation
The riddle states: remainder 1 when divided by 3, odd number (cannot be paired), remainder 1 when divided by 5, divisible by 7, and less than 100. Numbers divisible by 7 less than 100: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98. Odd numbers from these: 7, 21, 35, 49, 63, 77, 91. Those leaving remainder 1 when divided by 3: 7 (7÷3=2r1), 49 (49÷3=16r1), 91 (91÷3=30r1). Those leaving remainder 1 when divided by 5: only 91 (91÷5=18r1).
Solution Steps
Find numbers < 100 divisible by 7
Filter odd numbers: 7, 21, 35, 49, 63, 77, 91
Check remainder 1 when divided by 3: 7, 49, 91
Check remainder 1 when divided by 5: 91
Answer: 91 pebbles