Divisibility and the Division Algorithm · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

One quotient, one remainder

Mathematics · Number Theory · ages 18-19
Name ______________________   Date ____________

  1. What does a|b mean for integers a and b?

    • There is an integer c with b = a times c
    • b divided by a leaves remainder 1
    • a is bigger than b
    • a and b are both even

  2. Write 47 = 6q + r with 0 <= r < 6. What is r?

    Answer: ______________

  3. What does the statement a divides b mean?

    • That b divided by a leaves no quotient
    • That a and b share no common factor
    • That some integer c gives b = a times c

  4. In the division algorithm, the remainder r must satisfy 0 <= r < d.

    Circle one:   True   False

  5. If a|b and b|c, what follows directly from the definition?

    • a|c
    • c|a
    • a + c divides b
    • Nothing follows

  6. An even integer n is squared. What remainder does the square leave on division by 4?

    • 1
    • 3
    • 0

  7. What remainder does negative 16 leave on division by 26?

    • 10
    • negative 6
    • 16

  8. Write -23 = 5q + r with 0 <= r < 5. What is r?

    Answer: ______________

  9. Lee claims odd squares leave remainder 3 on division by 4. What is wrong?

    • Her remainder 3 never appears; 5 squared is 25, or 4 times 6 plus 1
    • Her quotient is wrong; 5 squared is 4 times 5 plus 5
    • She is right; odd squares leave remainder 3

  10. For any integer n, what remainder can n squared leave upon division by 4?

    • 0 or 1
    • 0, 1, 2 or 3
    • 2 or 3
    • Always 0
LightMySky · lightmysky.comW1-mt_UkQNs-6-40-s1

Answer key

For grown-ups. Fold this page away before handing over the rest.

One quotient, one remainder W1-mt_UkQNs-6-40-s1

  1. There is an integer c with b = a times c · Divisibility is defined by exact multiplication, never by example.
  2. 5 · 47 = 6 by 7 + 5, and 5 is below 6.
  3. That some integer c gives b = a times c · Divisibility is defined by that product, never by examples.
  4. True · The remainder must be zero or more and below the divisor.
  5. a|c · Write b = as and c = bt, then c = a(st), so a divides c.
  6. 0 · An even n is 2k, so its square is 4 times k squared, leaving 0.
  7. 10 · Negative 16 = 26 times negative 1 plus 10, and 10 qualifies.
  8. 2 · -23 = 5 by (-5) + 2, the unique nonnegative remainder.
  9. Her remainder 3 never appears; 5 squared is 25, or 4 times 6 plus 1 · Odd squares have the form 4 times something plus 1, never plus 3.
  10. 0 or 1 · Even n gives 0 mod 4; odd n = 2k+1 gives 4k(k+1)+1, remainder 1.
Worksheet · LightMySky