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Solved problems for the chapter:

Divisibility Rules

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Divisibility and division algorithm for KS4

1. Divisibility relation properties

1.1. Reflexivity

1.2. Antisymmetry

1.3. Transitivity

2. Divisibility relation in a set of integers

2.1. Basic theorem on division

3. Forum

↓ Show theory [1] ↓

Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:divisibility_rules/4149/

Example 1.

Find which of the numbers and is a multiple of all four numbers written:

Example 2.

Is a multiple of 17?

Find the minimum and maximum number which give a quotient of 7 when the divisor is 12.

Determine all numbers between 17000 and 35000 that are divisible by 5 and when divided by 1992 give a remainder of 104.

Using division theorem, write down the expression:

34 divided by 6
289 divided by 15

What are the possible remainders when a natural number is divided by 6?

Which natural number was divided by 7 to get a quotient of 8 and a remainder of 5?

Find the first 6 natural numbers, which when divided by 8 gives the remainder of 5.

Given that:

Prove that this value is divisible by 11.

10.

Find the multiples of 8 in the set of natural numbers. Write your answer in two ways.

11.

Prove that:

12.

Find the set of all natural numbers which when divided by 6 give remainder 2. Write your answers in two ways.

13.

Show that:

is divisible by 10.

14.

Find a number which:

when divided by 42, gives a remainder of 12.
divides 13 with a remainder. What's the remainder?

15.

Find an odd number that is divisible by 4 when 3 is added to its square.

16.

Given:

Prove that this expression is divisible by 7.

17.

Prove that:

18.

A number n divides a dividend to give a quotient of 7 and remainder 1. If also divides the dividend to give a quotient 5 and remainder of 3. Find n and the dividend.

19.

Prove that:

20.

The subtraction of 4 from the product of 7 and k equals the addition of 4 to the product of 6 and k. If k is a quotient, find k and the dividend.

21.

Prove that:

22.

Find the least number that gives a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5.

23.

Find the remainder when a three-digit number is divided by 7. If the sum of the digits of the three-digit number is 7 and the unit and tens digits are equal.

24.

There are different jobs available on a construction site with few workers at disposal. If five workers were assigned to each job, they would have two left, but if six workers were assigned to each job, they would need three workers more. Determine the number of workers available and how many different jobs they must perform.

25.

Find all the divisors of:

26.

Prove that:

there exist a two-digit number such that if zero is inserted between these two digits, the result will be a three-digit number divisible by 9 and the product of this two-digit number with 9 equals the three-digit number.

27.

Show that for , the following expression is divisible by 31:

28.

Show that:

29.

Prove that the sum of three consecutive odd powers of 2 are divisible by 3 and 7.

Find a number that divides the difference of the squares of two consecutive odd numbers, where the two consecutive odd numbers are and .

35.

If and are divisible by . Prove that is also divisible by .

36.

Prove that is divisible by 9 if is divisible by 6 and is a multiple of 5.

37.

Prove that:

If n is not divisible by 3, then its square reduced by 1 is divisible by 3.

38.

If is a perfect square, find the minimum even number which satisfies the expression.

39.

Prove that if n is a number, then is divisible by 4.

40.

At an event where chairs are to be arranged in rows, when 27 chairs are arranged in each row, 12 chairs are left and when 28 chairs are arranged in a row, there will be 5 chairs left. How many chairs are available for the event and how many possible rows can be gotten?