While exploring a remote mountain observatory, Emek and Tara discovered a series of numerical locks protecting decades of scientific research. Each lock displayed a different collection of integers and clues about divisibility, prime factors, remainders, and other number relationships. Instead of testing random combinations, they analyzed the mathematical patterns and verified that each deduction satisfied every condition before entering a code. As each lock opened, they gained access to another section of the archive. Their logical reasoning eventually unlocked the final vault, preserving the observatory's valuable scientific records for future researchers.
Challenges
Challenge 1 of 5
Challenges
Challenge 1 of 4
Hint
Check each choice for divisibility by both 12 and 15.
The first observatory lock opens with a number divisible by 12 and 15 but not divisible by 7. Which code satisfies all three conditions?
456084105Correct! 60 is divisible by both 12 and 15, but not by 7.
Check all three divisibility conditions before selecting the code.
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Hint
List numbers that leave remainder 2 when divided by 5, then test division by 7.
A numerical lock requires the smallest positive integer that leaves a remainder of 2 when divided by 5 and a remainder of 3 when divided by 7. What is the code?
Correct! 17 leaves remainder 2 when divided by 5 and remainder 3 when divided by 7.
Test the conditions one at a time and choose the smallest positive number satisfying both.
Next🎉 Congrats! You got this question!
Hint
Use only the prime factors shared by both numbers, with the smaller exponent for each.
The final lock displays the prime factorizations 360 = 2^3 × 3^2 × 5 and 168 = 2^3 × 3 × 7. Which value is the greatest common divisor of 360 and 168?
12182472Correct! The shared factors are 2^3 and 3, giving 24.
Compare the two prime factorizations and use the smaller exponent of each common prime.
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