During a mountain expedition, Emek and Tara followed a route protected by automated checkpoints that would activate only after the correct numerical puzzle was solved. Each checkpoint presented a different challenge involving divisibility, greatest common divisors, modular relationships, or other advanced properties of integers. Rather than using the same method every time, they selected the reasoning strategy that best matched each puzzle and verified every solution before proceeding. Their careful mathematical decisions activated every checkpoint in sequence, allowing the team to reach the summit safely and complete the expedition.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Find or test a common multiple of 18 and 24, then check divisibility by 5.
At the first checkpoint, the activation code must be divisible by 18 and 24 but not divisible by 5. Which code satisfies all three conditions?
487290120Correct! 72 is divisible by both 18 and 24, but not by 5.
Check every choice against all three divisibility conditions.
Next🎉 Congrats! You got this question!
Hint
Look for a repeating pattern in the remainders of powers of 3 when divided by 8.
A checkpoint asks for the remainder when 3^20 is divided by 8. What integer should Emek and Tara enter?
Correct! The remainder is 1.
Compute the first few powers of 3 modulo 8 and identify the cycle.
Next🎉 Congrats! You got this question!
Hint
Find the greatest common divisor of 84 and 126.
At the final checkpoint, 84 food packs and 126 water packs must be divided into the greatest possible number of identical supply groups with no items left over. How many groups can be made?
21284263Correct! The greatest common divisor is 42, so 42 identical groups can be made.
Use the greatest common divisor because the groups must be identical and leave no remainder.
Next🎉🎉🎉🎉🎉🎉🎉🎉
🎉 Congrats! You finished this challenge set!
