At the peak of an ancient mountain sanctuary, Emek and Tara discovered a vault protected by a network of interconnected numerical clues. Each puzzle depended on properties of integers, divisibility, modular arithmetic, and logical relationships established by earlier solutions. Several approaches appeared promising, but they carefully compared competing arguments, tested each conclusion against every condition, and rejected any result that created a contradiction. Only one sequence of deductions remained fully consistent. Their rigorous mathematical reasoning unlocked the vault, revealing a priceless collection of historical manuscripts that had been preserved for centuries.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
First list the multiples of 6 in the given interval, then check their remainders when divided by 5.
One vault clue states that the code is divisible by 6, leaves a remainder of 1 when divided by 5, and is between 30 and 60. Which code satisfies every condition?
36424854Correct! 36 is divisible by 6 and leaves a remainder of 1 when divided by 5.
Test each choice against both the divisibility and remainder conditions.
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Hint
List numbers congruent to 2 modulo 7, then test which one leaves remainder 4 when divided by 9.
A later clue requires the smallest positive integer that leaves a remainder of 2 when divided by 7 and a remainder of 4 when divided by 9. What code should Emek and Tara enter?
Correct! 58 satisfies both remainder conditions.
Check the numbers that leave remainder 2 when divided by 7 until both conditions are satisfied.
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Hint
Look for a counterexample that is divisible by both 4 and 6.
Tara argues, "If an integer is divisible by 4 and by 6, then it must be divisible by 24." Which evaluation of her argument is correct?
The argument is correct because 4 × 6 = 24.The argument is incorrect because such an integer must be divisible by 10.The argument is incorrect because 12 is divisible by both 4 and 6 but not by 24.The argument is correct because every multiple of 4 is also a multiple of 24.Correct! The number 12 disproves the claim.
A single counterexample is enough to show that the statement is false.
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