Divisibility and Remainder Reasoning
A remainder is what is left after exact grouping, and it must stay smaller than the divisor. Problems range from finding common multiples and numbers meeting divisibility clues, to recovering a dividend from quotient-and-remainder, adjusting a total to leave no remainder, rounding bundles up, and maximizing a remainder. The controlling fact is the division algorithm: dividend = divisor x quotient + remainder.
grade 3–4 Measurement & dataBase-ten numbersOperations Eliminate PossibilitiesMake a Systematic List
Progression (8)
g3 1. Numbers divisible by several are common multiples Operations
g3 2. Narrow candidates by divisibility conditions Operations
g3 3. Adjust the total to leave no remainder Operations
g3 4. Exact division means remainder zero Operations
g3 5. Recover the dividend from quotient and remainder Operations
g3 6. Remainder must be less than the divisor Operations
g3 7. Round the quotient up to carry the remainder OperationsMeasurement & data
g4 8. The remainder is always less than the divisor Base-ten numbers
Bridges to competition (5)
Reasoning problems that extend this idea toward competition:
Build division expressions meeting conditions Number arrangement
Possible remainder values Number arrangement
Capstone: grid-sum and remainder conditions Logic
Find a number from division clues Logic
Smallest number meeting remainder bounds Logic