MikeFromHC

Even today with calculators and computers there are times when it is helpful to see if one number will divide into another.

There are simple rules for 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12 and any number which is a form of 2^n.

Seven is the exception and while there are some rules it is easier to just divide the thing.

I'll leave one to you

Two

Two itself is trivial but follows the rule. If the last n digits are divisible by 2^n, the number is divisible by 2^n2 is 2^1, 4 is 2^2, 8 is 2^3...

All are forms of 2^n

If the last digit is even, the number can be divided by 2

If the last two digits are divisible by 4 (2^2) then the number can be divided by 4

If the last three digits are divisible by 8 (2^3) then the number can be divided by 8

etc.

Three

Sum the digits until you have a single digit or see the answer.

If the sum is divisible by three the number is also. (See below for a bit more)

12345 = 1+2+3+4+5= 15 = 1+5 = 6

The number can be divided by 3.

Four

A form of 2^n If the last two digits are divisible then the number is.

Five

If it ends in five or zero it is divisible

Six

If it is even and divisible by three it is divisible by six

Seven

Just do the math.

Eight

A form of 2^n If the last three digits are divisible then the number is.

Nine

Same as three, sum the digits and if the result is 9 it is divisible.(See below for a bit more)

Ten

If it ends in zero it is divisible.

Note that this also follows the 10^n rule.

Eleven

First digit minus second digit plus third digit minus...

If the answer is eleven or zero the number is devisable by eleven.

65*11 equals 715

7-1+5 = 11

99

9-9 = 0

Twelve

If divisible by three and four it can be divided by twelve.

Below The rule for nine and three is known as "casting out" and especially for nine has many other uses.

Summing the digits is not always needed as you can "cast out" sets of numbers.

For three

12345 = 1+2+3+4+5

Note that 1+2 = 3 so you can just omit them, the 3 itself, and just look at 45.

For nine

1234512 = 1+2+3+4+5 + 1 + 2

4+5 = 9 so I am left with 12312 which sums to 9.

NOTE If the sum of the digits is not equal to 9 then that number is the remainder.

Final comment. If anyone is interested in how useful casting out nines is, let me know.

There are simple rules for 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12 and any number which is a form of 2^n.

Seven is the exception and while there are some rules it is easier to just divide the thing.

I'll leave one to you

Two

Two itself is trivial but follows the rule. If the last n digits are divisible by 2^n, the number is divisible by 2^n2 is 2^1, 4 is 2^2, 8 is 2^3...

All are forms of 2^n

If the last digit is even, the number can be divided by 2

If the last two digits are divisible by 4 (2^2) then the number can be divided by 4

If the last three digits are divisible by 8 (2^3) then the number can be divided by 8

etc.

Three

Sum the digits until you have a single digit or see the answer.

If the sum is divisible by three the number is also. (See below for a bit more)

12345 = 1+2+3+4+5= 15 = 1+5 = 6

The number can be divided by 3.

Four

A form of 2^n If the last two digits are divisible then the number is.

Five

If it ends in five or zero it is divisible

Six

If it is even and divisible by three it is divisible by six

Seven

Just do the math.

Eight

A form of 2^n If the last three digits are divisible then the number is.

Nine

Same as three, sum the digits and if the result is 9 it is divisible.(See below for a bit more)

Ten

If it ends in zero it is divisible.

Note that this also follows the 10^n rule.

Eleven

First digit minus second digit plus third digit minus...

If the answer is eleven or zero the number is devisable by eleven.

65*11 equals 715

7-1+5 = 11

99

9-9 = 0

Twelve

If divisible by three and four it can be divided by twelve.

Below The rule for nine and three is known as "casting out" and especially for nine has many other uses.

Summing the digits is not always needed as you can "cast out" sets of numbers.

For three

12345 = 1+2+3+4+5

Note that 1+2 = 3 so you can just omit them, the 3 itself, and just look at 45.

For nine

1234512 = 1+2+3+4+5 + 1 + 2

4+5 = 9 so I am left with 12312 which sums to 9.

NOTE If the sum of the digits is not equal to 9 then that number is the remainder.

Final comment. If anyone is interested in how useful casting out nines is, let me know.