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 3–6 Measurement & dataBase-ten numbersOperations Eliminate PossibilitiesMake a Systematic List
Progression (20)
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
g4 9. Count equal-share ways using divisors Operations
g4 10. A multiple is divisible by its factor Operations
g4 11. Knowing the multiple aids factoring into products Operations
g6 12. Product of two numbers equals GCF times LCM Number system
g4 13. Subtract the remainder to find a divisor OperationsBase-ten numbers
g4 14. Common factor over 1 means not in lowest terms OperationsFractions
g4 15. Choose round up or round down in context OperationsBase-ten numbers
g6 16. A denominator that cancels to 1 leaves a whole number Number system
g6 17. Subtract the remainder and the division comes out exact Number system
g6 18. A denominator that cancels to 1 leaves a whole number Number system
g6 19. Rebuild the dividend with multiplication and addition Number system
g6 20. To divide exactly, drop the leftover or top it up Number system
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