Bundle Remainders
A school is packing exam booklets into bundles for storage. When the booklets are packed in bundles of 9, 4 booklets are left over. When they are packed in bundles of 12, 7 booklets are left over. The total number of booklets is known to be between 300 and 500. The caretaker also remembers that when the booklets were stacked in piles of 5, exactly 2 were left over. A teacher suggests checking multiples of 9 and 12 separately, but the student council wants a systematic way to list all numbers satisfying the first two conditions before applying the third. They must also decide whether the answer is unique or whether more information is needed. The booklets are identical, and no bundle may be partially filled except for the stated remainders.
1. Find all integers between 300 and 500 satisfying the first two remainder conditions. Show a systematic listing or congruence argument.
Read the worked answer
Write n = 4 + 9k. The second condition gives 4 + 9k ≡ 7 (mod 12), so 9k ≡ 3 (mod 12). Dividing by 3 (valid because the modulus also divides by 3) gives 3k ≡ 1 (mod 4). Since 3·3 ≡ 1 (mod 4), k ≡ 3 (mod 4). Thus k = 3 + 4t and n = 4 + 9(3+4t) = 31 + 36t. The range 300 ≤ n ≤ 500 gives 300 ≤ 31+36t ≤ 500, so 269 ≤ 36t ≤ 469, hence t = 8,9,10,11,12,13. The candidates are 319, 355, 391, 427, 463, 499. Checking one, 319 = 9·35 + 4 and 319 = 12·26 + 7.
2. Apply the third condition and state the total number of booklets.
Read the worked answer
Now require n ≡ 2 (mod 5). Check the candidates: 319 ≡ 4, 355 ≡ 0, 391 ≡ 1, 427 ≡ 2, 463 ≡ 3, 499 ≡ 4 (mod 5). The only match is n = 427. Verification: 427 = 9·47 + 4, 427 = 12·35 + 7, and 427 = 5·85 + 2. Therefore the total is 427 booklets.
3. Explain why a student who only checks 427 against the first two conditions has not proved uniqueness. What if the third condition were 'remainder 3 when divided by 5' instead?
Read the worked answer
Checking only 427 against the first two conditions shows that it is a solution, but it does not rule out 319, 355, 391, 463, or 499, which also satisfy those first two conditions. Uniqueness comes from imposing the third condition on the complete candidate list. If the third condition were 'remainder 3 when divided by 5', then 463 would be the unique match, because it is the only candidate congruent to 3 modulo 5. The method is therefore: generate all solutions to the first two conditions, then filter by the third; do not stop at the first number that works.