Applying Divisibility Proofs In Problems

1) If a number leaves remainder 1 when divided by 3, which equation represents it?

N = 3k

N = 3k + 1

N = 3k + 2

N = 3k − 1

When a number is divided by 3 and leaves a remainder of 1, it can be expressed as 3k + 1 because it is 1 more than a multiple of 3.

Explanation

When a number is divided by 3 and leaves a remainder of 1, it can be expressed as 3k + 1 because it is 1 more than a multiple of 3.

2) Which value of x makes 5x + 2 divisible by 3?

X = 1

X = 2

X = 4

X = 6

To make 5x + 2 divisible by 3, substitute values: if x = 2, then 5(2) + 2 = 12, which divides evenly by 3.

Explanation

To make 5x + 2 divisible by 3, substitute values: if x = 2, then 5(2) + 2 = 12, which divides evenly by 3.

3) Which number satisfies both: divisible by 6 and leaves remainder 2 when divided by 5?

10

22

32

42

For a number that’s divisible by 6 and leaves remainder 2 when divided by 5, 42 works because 42 ÷ 6 = 7 and 42 ÷ 5 leaves a remainder of 2.

Explanation

For a number that’s divisible by 6 and leaves remainder 2 when divided by 5, 42 works because 42 ÷ 6 = 7 and 42 ÷ 5 leaves a remainder of 2.

4) Solve for x: 2x + 3 is divisible by 5. Which x works?

X = 1

X = 2

X = 3

X = 4

If 2x + 3 is divisible by 5, substitute x = 1: 2(1) + 3 = 5, which divides evenly by 5.

Explanation

If 2x + 3 is divisible by 5, substitute x = 1: 2(1) + 3 = 5, which divides evenly by 5.

5) Which number leaves remainder 4 when divided by both 5 and 7?

24

39

49

74

A number that leaves remainder 4 when divided by both 5 and 7 is 39 because 39 ÷ 5 and 39 ÷ 7 each leave a remainder of 4.

Explanation

A number that leaves remainder 4 when divided by both 5 and 7 is 39 because 39 ÷ 5 and 39 ÷ 7 each leave a remainder of 4.

6) If x is an even number and 3x + 1 is divisible by 5, which of these could x be?

4

6

8

10

If x is even and 3x + 1 is divisible by 5, try x = 8: 3(8) + 1 = 25, which divides by 5. 8 is even, so it satisfies both conditions.

Explanation

If x is even and 3x + 1 is divisible by 5, try x = 8: 3(8) + 1 = 25, which divides by 5. 8 is even, so it satisfies both conditions.

7) Which of the following equations represents numbers that are divisible by both 2 and 3 but leave remainder 1 when divided by 5?

N = 5k

N = 6k + 5

N = 5k + 1

N = 30k + 6

Numbers divisible by both 2 and 3 but leaving remainder 1 when divided by 5 can be expressed using their LCM (30): 30k + 6 gives the needed pattern.

Explanation

Numbers divisible by both 2 and 3 but leaving remainder 1 when divided by 5 can be expressed using their LCM (30): 30k + 6 gives the needed pattern.

8) Which number is divisible by 6 but not by 9?

18

24

36

45

A number divisible by 6 but not by 9 is 24 because 24 ÷ 6 = 4, yet 24 is not a multiple of 9.

Explanation

A number divisible by 6 but not by 9 is 24 because 24 ÷ 6 = 4, yet 24 is not a multiple of 9.

9) A number divided by 8 leaves remainder 3. Which equation models this?

N = 8k

N = 8k + 1

N = 8k + 3

N = 8k + 5

When a number is divided by 8 and leaves a remainder of 3, it can be written as 8k + 3.

Explanation

When a number is divided by 8 and leaves a remainder of 3, it can be written as 8k + 3.

10) Which number is divisible by 5 and leaves remainder 2 when divided by 3?

10

15

20

25

A number divisible by 5 and leaving a remainder of 2 when divided by 3 is 20 because 20 ÷ 5 = 4 and 20 ÷ 3 leaves remainder 2.

Explanation

A number divisible by 5 and leaving a remainder of 2 when divided by 3 is 20 because 20 ÷ 5 = 4 and 20 ÷ 3 leaves remainder 2.

11) A number is divisible by both 5 and 9. Which equation represents it?

N = 5k + 9

N = 9k + 5

N = 10k

N = 45k

A number divisible by both 5 and 9 must be a multiple of their LCM, 45, so it can be written as 45k.

Explanation

A number divisible by both 5 and 9 must be a multiple of their LCM, 45, so it can be written as 45k.

12) A number is divisible by 4 and leaves remainder 1 when divided by 3. Which equation represents it?

N = 3k + 1

N = 4k

N = 4k + 3

N = 12k + 4

A number divisible by 4 and leaving remainder 1 when divided by 3 fits the pattern 12k + 4 because 12 is the least common multiple of 4 and 3, and adding 4 creates the required remainder.

Explanation

A number divisible by 4 and leaving remainder 1 when divided by 3 fits the pattern 12k + 4 because 12 is the least common multiple of 4 and 3, and adding 4 creates the required remainder.

13) Solve for x: 4x − 7 = 3x + 5

X = 7

X = 12

X = 15

X = 20

Solving 4x − 7 = 3x + 5 gives x = 12 after adding 7 and subtracting 3x.

Explanation

Solving 4x − 7 = 3x + 5 gives x = 12 after adding 7 and subtracting 3x.

14) Find the smallest number greater than 50 that is divisible by both 4 and 6.

52

54

56

60

The smallest number greater than 50 that is divisible by both 4 and 6 is 60 because the LCM of 4 and 6 is 12, and the next multiple of 12 after 50 is 60.

Explanation

The smallest number greater than 50 that is divisible by both 4 and 6 is 60 because the LCM of 4 and 6 is 12, and the next multiple of 12 after 50 is 60.