Answer to Question #350466 in Combinatorics | Number Theory for Lefa

Question

Test the following numbers for divisibility by 6,9,11 (do not divide or factorise)

a)6 798 340

b)54 786 978

Answer 1

If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.

If the even number is divisible by 3, then it is divisible by 6.

If the sum of the digits of a number is divisible by 9, then the number is divisible by 9.

if the difference of the sum of the digits at even places and odd places is 0 or divisible by 11, then the number is divisible by 11.

a)

The number $6 798 340$ is an even number.

The sum of the digits is $6+7+9+8+3+4=37.$

The number $6 798 340$ is neither divisible by 6 nor by 9.

The difference of the sum of the digits at even places and odd places is

$$7+8+4-(6+9+3+0)=1$$

The number $6 798 340$ is not divisible by 11.

b)

The number $54 786 978$ is an even number.

The sum of the digits is $5+4+7+8+6+9+7+8=54.$

The number $54 786 978$ is divisible by 6.

The number $54 786 978$ is divisible by 9.

The difference of the sum of the digits at even places and odd places is

$$4+8+9+8-(5+7+6+7)=4$$

The number $54 786 978$ is not divisible by 11.