What changes when a number must be an integer?
Arithmetic is part of the Calculation content area described by FNEGE. The exercises that follow are educational examples created by Upsilar. They focus on expressing constraints; advanced formalism is not needed to determine whether items can be divided into full groups.
For integers n and d, where d is positive, Euclidean division is written n = d × q + r, where q and r are integers and 0 ≤ r < d. The remainder must be less than the divisor. If you write a remainder of 9 when dividing by 7, you can still form another group of 7, so the quotient and remainder are not final.
Next, underline the permitted domain. “Nonnegative integer,” “digit,” and “two-digit number” are not interchangeable. An equation may allow a decimal value that the context rules out. Conversely, do not add an integer requirement when the problem does not state one.
| Clue | Useful expression |
|---|---|
| n is even | n = 2k for an integer k |
| n is a multiple of 5 | n = 5k for an integer k |
| Remainder 3 when divided by 7 | n = 7k + 3 for an integer k |
| Unknown ones digit | A value from 0, 1, …, 9 |
How can you combine simple divisibility rules?
For divisibility by 2, check the ones digit; for 3 and 9, add the digits; for 5, check whether the number ends in 0 or 5. Divisibility by 6 means divisibility by both 2 and 3. These checks apply to an exact integer, not a rounded number.
Quel chiffre peut remplacer x dans le nombre 4x2 pour que ce nombre soit divisible par 9 ?
Choose an answer.
The solution, step by step
x = 3.
The method in action
- The digit sum is 4 + x + 2 = 6 + x. Since x ranges from 0 to 9, the sum is between 6 and 15.
- The only multiple of 9 in this range is 9, so x = 3. Indeed, 432 = 9 × 48.
- With x = 6, the sum would be 12: the number would be divisible by 3, but not by 9. x = 9 does not work merely because the digit 9 appears in the number.
What is the remainder after a transformation?
A form such as n = 7k + 3 lets you transform the entire number without knowing k. At the end, separate out a multiple of the divisor, then reduce the remainder if it is too large.
Un entier n laisse un reste de 3 lorsqu’on le divise par 7. Quel est le reste de la division de 2n + 5 par 7 ?
Choose an answer.
The solution, step by step
4.
The method in action
- Write n = 7k + 3. Then 2n + 5 = 14k + 11 = 7(2k + 1) + 4.
- The remainder is 4, which is between 0 and 6. For n = 10, 2n + 5 = 25, and 25 = 3 × 7 + 4.
- 11 is an intermediate value, not a valid remainder modulo 7. Keeping 3 would ignore both operations applied to n.
The pitfall
Multiplying two divisors that already share a factor
“Divisible by 4 and by 6” does not imply “divisible by 24.” The number 12 is an immediate counterexample. The two constraints share a factor of 2; together, they require a multiple of 12. One counterexample is enough to disprove a universal rule, even if many other numbers satisfy it.
Likewise, two odd numbers have an even sum, but their product is still odd. A rule about a sum does not carry over to a product. Write down the operation before reasoning about parity.
How can you find the possible values without testing every integer?
First list the more restrictive set, then apply the second constraint.
Un nombre entier est strictement compris entre 30 et 50. Il est divisible par 4 et par 6. Quelles sont toutes ses valeurs possibles ?
Choose an answer.
The solution, step by step
36 and 48.
The method in action
- The multiples of 12 strictly between 30 and 50 are 36 and 48. Both are divisible by 4 and by 6.
- 32 is divisible by 4 but not by 6. Choices that include only 36 or only 48 miss a valid possibility.
- To force a single value, you could add a condition such as “less than 40.” The only remaining value would then be 36. Finding the first number that works does not prove it is unique.
Key takeaways
- Name the domain and bounds of the unknowns.
- A remainder is a nonnegative integer strictly less than the positive divisor.
- Check for uniqueness and look for counterexamples to assumed rules.