Algebraic Proofs Of Divisibility

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1) A number leaves a remainder of 2 when divided by 5. Which equation represents this situation?

N = 5k

N = 5k + 1

N = 5k + 2

N = 5k - 2

When a number n is divided by 5 and leaves a remainder of 2, it can be expressed in the form of 5k + 2, where k is an integer. This means that n is 2 more than a multiple of 5, which corresponds to option C.

Explanation

About This Quiz

Numbers follow patterns, and algebra is the key to proving them! In this quiz, you’ll use algebraic expressions like n = 2k and n = 3k to show when numbers are divisible by 2, 3, 5, 9, and more. You’ll also explore how equations can represent all even, odd, o... see moremultiple-based numbers. We bring you this quiz to strengthen your proof skills and connect arithmetic rules with algebra.
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2) A number n is divisible by 4. Which equation represents this?

N = 2k

N = 3k

N = 4k

N = 5k

If n is divisible by 4, it can be expressed as 4k because it is exactly 4 times some integer k with no remainder.

Explanation

If n is divisible by 4, it can be expressed as 4k because it is exactly 4 times some integer k with no remainder.

3) A number is divisible by 12 if it is divisible by:

2 and 5

3 and 9

3 and 4

4 and 5

A number is divisible by 12 if it is divisible by both 3 and 4, since 12 = 3 × 4.

Explanation

A number is divisible by 12 if it is divisible by both 3 and 4, since 12 = 3 × 4.

4) Which algebraic form shows all even numbers?

X = 2n + 1

X = 3n

X = 4n

X = 2n

Even numbers can be represented as 2n because they are multiples of 2.

Explanation

Even numbers can be represented as 2n because they are multiples of 2.

5) Which equation models all numbers divisible by both 2 and 5?

X = 2n

X = 5n

X = 10n

X = 7n

Numbers divisible by both 2 and 5 must be multiples of their least common multiple, 10, so they can be written as 10n.

Explanation

Numbers divisible by both 2 and 5 must be multiples of their least common multiple, 10, so they can be written as 10n.

6) A number divided by 7 leaves a remainder of 4. Which equation models this?

X = 7k + 1

X = 7k + 2

X = 7k + 4

X = 7k − 4

When a number is divided by 7 and leaves a remainder of 4, it can be expressed as 7k + 4, meaning it is 4 more than a multiple of 7.

Explanation

When a number is divided by 7 and leaves a remainder of 4, it can be expressed as 7k + 4, meaning it is 4 more than a multiple of 7.

7) Solve for x: 3(x + 4) = 2(x + 7)

X = 2

X = 6

X = 7

X = 8

Solving 3(x + 4) = 2(x + 7): expand to 3x + 12 = 2x + 14, so x = 2.

Explanation

Solving 3(x + 4) = 2(x + 7): expand to 3x + 12 = 2x + 14, so x = 2.

8) Solve for x: 4x + 5 = 2x + 15

X = 2

X = 5

X = 7

X = 10

Solving 4x + 5 = 2x + 15: subtract 2x → 2x = 10 → x = 5.

Explanation

Solving 4x + 5 = 2x + 15: subtract 2x → 2x = 10 → x = 5.

9) Solve for x: 5(x − 3) = 3(x − 1)

X = 0

X = 5

X = 6

X = 7

Solving 5(x − 3) = 3(x − 1): expand → 5x − 15 = 3x − 3 → 2x = 12 → x = 6.

Explanation

Solving 5(x − 3) = 3(x − 1): expand → 5x − 15 = 3x − 3 → 2x = 12 → x = 6.

10) Solve for x: 3(x + 2) = 2(x + 8)

X = 4

X = 8

X = 10

X = 12

Solving 3(x + 2) = 2(x + 8): expand → 3x + 6 = 2x + 16 → x = 10.

Explanation

Solving 3(x + 2) = 2(x + 8): expand → 3x + 6 = 2x + 16 → x = 10.

11) Solve for x: 2x + 5 = x + 11

X = 3

X = 4

X = 5

X = 6

Solving 2x + 5 = x + 11: subtract x → x = 6.

Explanation

Solving 2x + 5 = x + 11: subtract x → x = 6.

12) Which equation represents all numbers divisible by 9?

N = 9k + 1

N = 9k + 2

N = 9k

N = 9k − 1

Numbers divisible by 9 can be written as 9 times an integer, so n = 9k.

Explanation

Numbers divisible by 9 can be written as 9 times an integer, so n = 9k.

13) Which algebraic form shows all odd numbers?

N = 2k

N = 2k + 1

N = 3k

N = 5k

Odd numbers are one more than even numbers, so they take the form 2k + 1.

Explanation

Odd numbers are one more than even numbers, so they take the form 2k + 1.

14) Which equation models all numbers divisible by both 3 and 5?

X = 5n

X = 3n

X = 15n

X = 7n

Numbers divisible by both 3 and 5 are multiples of 15, so they can be written as 15n.

Explanation

Numbers divisible by both 3 and 5 are multiples of 15, so they can be written as 15n.

15) If n = 6k, then n is always:

Divisible by 4

Divisible by 5

Divisible by 6

Divisible by 7

If n = 6k, it is a multiple of 6 and therefore divisible by 6.

Explanation

If n = 6k, it is a multiple of 6 and therefore divisible by 6.

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Cierra Henderson |MBA |

K-12 Expert

Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.

Algebraic Proofs of Divisibility