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Modulo worksheet

Grades 4–6 · CCSS 4.NBT.B.6, 5.NBT.B.6

Solve each problem and show your work. Round answers to 2 decimal places where needed.

1.
Problem 1.
- Dividend (a): 18
- Divisor, or modulus (n): 5
- Convention for negative numbers: Floored (sign of n)
Find: a mod n (remainder)
Answer
2.
Problem 2.
- Dividend (a): 15
- Divisor, or modulus (n): 4
- Convention for negative numbers: Euclidean (never negative)
Find: a mod n (remainder)
Answer
3.
Problem 3.
- Dividend (a): 16
- Divisor, or modulus (n): 6
- Convention for negative numbers: Floored (sign of n)
Find: a mod n (remainder)
Answer
4.
Problem 4.
- Dividend (a): 20
- Divisor, or modulus (n): 6
- Convention for negative numbers: Truncated (sign of a)
Find: a mod n (remainder)
Answer
5.
Problem 5.
- Dividend (a): 15
- Divisor, or modulus (n): 4
- Convention for negative numbers: Truncated (sign of a)
Find: a mod n (remainder)
Answer
6.
Problem 6.
- Dividend (a): 14
- Divisor, or modulus (n): 6
- Convention for negative numbers: Truncated (sign of a)
Find: a mod n (remainder)
Answer
7.
Problem 7.
- Dividend (a): 24
- Divisor, or modulus (n): 7
- Convention for negative numbers: Truncated (sign of a)
Find: a mod n (remainder)
Answer
8.
Problem 8.
- Dividend (a): 16
- Divisor, or modulus (n): 4
- Convention for negative numbers: Euclidean (never negative)
Find: a mod n (remainder)
Answer
9.
Problem 9.
- Dividend (a): 16
- Divisor, or modulus (n): 4
- Convention for negative numbers: Floored (sign of n)
Find: a mod n (remainder)
Answer
10.
Problem 10.
- Dividend (a): 16
- Divisor, or modulus (n): 3
- Convention for negative numbers: Truncated (sign of a)
Find: a mod n (remainder)
Answer

Answer key: Modulo worksheet

1. a mod n (remainder): 3
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{18}{5}$ $= 3.6$
3. Find the whole number of times n fits (q). Floored rule: always round down, even below zero, so the remainder gets the sign of n
  $q$ $= ⌊ 3.6 ⌋$ $= 3$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 18 - 3 \times 5$ $= 3$
2. a mod n (remainder): 3
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{15}{4}$ $= 3.75$
3. Find the whole number of times n fits (q). Euclidean rule: pick q so the remainder is never negative. Round a ÷ |n| down (|n| is n without its sign), then give q the sign of n
  $q$ $= sign(n) × ⌊ 3.75 ⌋$ $= 3$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 15 - 3 \times 4$ $= 3$
3. a mod n (remainder): 4
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{16}{6}$ $= 2.6666666667$
3. Find the whole number of times n fits (q). Floored rule: always round down, even below zero, so the remainder gets the sign of n
  $q$ $= ⌊ 2.6666666667 ⌋$ $= 2$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 16 - 2 \times 6$ $= 4$
4. a mod n (remainder): 2
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{20}{6}$ $= 3.3333333333$
3. Find the whole number of times n fits (q). Truncated rule: chop off the decimal part, rounding toward zero, so the remainder gets the sign of a
  $q$ $= trunc (3.3333333333)$ $= 3$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 20 - 3 \times 6$ $= 2$
5. a mod n (remainder): 3
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{15}{4}$ $= 3.75$
3. Find the whole number of times n fits (q). Truncated rule: chop off the decimal part, rounding toward zero, so the remainder gets the sign of a
  $q$ $= trunc (3.75)$ $= 3$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 15 - 3 \times 4$ $= 3$
6. a mod n (remainder): 2
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{14}{6}$ $= 2.3333333333$
3. Find the whole number of times n fits (q). Truncated rule: chop off the decimal part, rounding toward zero, so the remainder gets the sign of a
  $q$ $= trunc (2.3333333333)$ $= 2$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 14 - 2 \times 6$ $= 2$
7. a mod n (remainder): 3
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{24}{7}$ $= 3.4285714286$
3. Find the whole number of times n fits (q). Truncated rule: chop off the decimal part, rounding toward zero, so the remainder gets the sign of a
  $q$ $= trunc (3.4285714286)$ $= 3$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 24 - 3 \times 7$ $= 3$
8. a mod n (remainder): 0
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{16}{4}$ $= 4$
3. Find the whole number of times n fits (q). Euclidean rule: pick q so the remainder is never negative. Round a ÷ |n| down (|n| is n without its sign), then give q the sign of n
  $q$ $= sign(n) × ⌊ 4 ⌋$ $= 4$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 16 - 4 \times 4$ $= 0$
9. a mod n (remainder): 0
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{16}{4}$ $= 4$
3. Find the whole number of times n fits (q). Floored rule: always round down, even below zero, so the remainder gets the sign of n
  $q$ $= ⌊ 4 ⌋$ $= 4$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 16 - 4 \times 4$ $= 0$
10. a mod n (remainder): 1
1. Write a as a whole number (q) of n's plus a leftover remainder (r). That leftover is the answer
  $a$ $= q \times n + r$
2. Divide a by n to see roughly how many times n fits into a
  $\frac{16}{3}$ $= 5.3333333333$
3. Find the whole number of times n fits (q). Truncated rule: chop off the decimal part, rounding toward zero, so the remainder gets the sign of a
  $q$ $= trunc (5.3333333333)$ $= 5$
4. Find the remainder r: take q lots of n away from a, and what's left is the answer
  $r$ $= 16 - 5 \times 3$ $= 1$