A library has 780 English and 364 Science books. Each shelf must have the same number of books of each subject. What is the minimum number of books on each shelf?
104 books per shelf (52 of each subject)
Idea: The number of books of each subject on one shelf must divide both 780 and 364. Use the greatest common divisor (GCD).
Find GCD(780, 364) using Euclid’s division:
First step:
\(780 \div 364 = 2\)
Remainder:
\(780 - 364\times 2 = 52\)
Second step:
\(364 \div 52 = 7\)
Remainder:
\(364 - 52\times 7 = 0\)
When the remainder becomes 0, the last non-zero remainder is the GCD.
So, \(\gcd(780,\,364) = 52\).
Use the GCD:
English books per shelf = \(52\)
Science books per shelf = \(52\)
Total per shelf = \(52 + 52 = 104\)
Conclusion: Minimum books on each shelf = 104 (52 of each subject).