Divisibility Rules And Algebraic Forms

1) A number is divisible by 2 if:

It ends in 1

It ends in 5

It ends in 0, 2, 4, 6, or 8

Its digits add to 3

Even numbers are multiples of 2. A number’s divisibility by 2 depends only on its ones digit.

Explanation

Even numbers are multiples of 2. A number’s divisibility by 2 depends only on its ones digit.

2) Which number confirms the divisibility rule for 3 (digit sum divisible by 3)?

25

123

124

202

If the sum of digits is a multiple of 3, so is the whole number. 1+2+3=6, a multiple of 3.

Explanation

If the sum of digits is a multiple of 3, so is the whole number. 1+2+3=6, a multiple of 3.

3) A number is divisible by 9 if:

The sum of its digits is divisible by 9

Its last two digits are divisible by 4

It ends in 0 or 5

It has three factors

Because 10 ≡ 1 (mod 9), so each place value acts like “1,” meaning the digit sum determines divisibility by 9.

Explanation

Because 10 ≡ 1 (mod 9), so each place value acts like “1,” meaning the digit sum determines divisibility by 9.

4) If a number ends in 0, it is divisible by:

2

3

7

10

Ending in 0 means it’s a multiple of both 2 and 5 — hence divisible by 10.

Explanation

Ending in 0 means it’s a multiple of both 2 and 5 — hence divisible by 10.

5) 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

The number is two more than a multiple of 5.

Explanation

The number is two more than a multiple of 5.

6) A number n is divisible by 4. Which equation represents this?

N = 2k

N = 3k

N = 4k

N = 5k

Multiplying any integer by 4 produces a multiple of 4.

Explanation

Multiplying any integer by 4 produces a multiple of 4.

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

2 and 5

3 and 9

3 and 4

4 and 5

The least common multiple (LCM) of 3 and 4 is 12.

Explanation

The least common multiple (LCM) of 3 and 4 is 12.

8) Which algebraic form shows all even numbers?

X = 2n + 1

X = 3n

X = 4n

X = 2n

Multiplying any integer by 2 gives an even number.

Explanation

Multiplying any integer by 2 gives an even number.

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

X = 2n

X = 5n

X = 10n

X = 7n

The smallest number divisible by 2 and 5 is 10 → multiples of 10 work for both.

Explanation

The smallest number divisible by 2 and 5 is 10 → multiples of 10 work for both.

10) 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

Every such number is 4 more than a multiple of 7.

Explanation

Every such number is 4 more than a multiple of 7.

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

X = 2

X = 6

X = 7

X = 8

3x + 12 = 2x + 14 → x = 2. Linear equation — isolate x.

Explanation

3x + 12 = 2x + 14 → x = 2. Linear equation — isolate x.

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

X = 2

X = 5

X = 7

X = 10

Subtract 2x → 2x = 10 → x = 5.

Explanation

Subtract 2x → 2x = 10 → x = 5.

13) Solve for x: 5(x − 3) = 3(x − 1)

X = 0

X = 5

X = 6

X = 7

5x – 15 = 3x – 3 → 2x = 12 → x = 6.

Explanation

5x – 15 = 3x – 3 → 2x = 12 → x = 6.

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

X = 4

X = 8

X = 10

X = 12

3x + 6 = 2x + 16 → x = 10.

Explanation

3x + 6 = 2x + 16 → x = 10.

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

X = 3

X = 4

X = 5

X = 6

Move x → x = 6.

Explanation

Move x → x = 6.

Divisibility Rules and Algebraic Forms

1) A number is divisible by 2 if:

2) Which number confirms the divisibility rule for 3 (digit sum divisible by 3)?

3) A number is divisible by 9 if:

4) If a number ends in 0, it is divisible by:

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

6) A number n is divisible by 4. Which equation represents this?

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

8) Which algebraic form shows all even numbers?

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

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

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

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

13) Solve for x: 5(x − 3) = 3(x − 1)

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

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