[Multiple of 2]

If the last figure of a number is even, then the number is a multiple of 2.

[Multiple of 3]

If the sum of every figure is a multiple of 3, then the number is a multiple of 3.

(Example)

8262; because 8+2+6+2=18 is a multiple of 3, 8362 is a multiple of 3.

(Reason)

8262=8000+200+60+2

=8*(999+1)+2*(99+1)+6*(9+1)+2

=(8*999+2*99+6*9)+(8+2+6+2)

What is firstly parenthesized is a multiple of 3, so you only have to check (8+2+6+2).

[Multiple of 4]

If the last two figures of a number is a multiple of 4, then the number
is a multiple of 4.

[Multiple of 5]

If the last figure of a number is 0 or 5, then the number is a multiple
of 5.

[Multiple of 9]

If the sum of every figure is a multiple of 9, then the number is a multiple of 9.

(Example)

8262; because 8+2+6+2=18 is a multiple of 9, 8362 is a multiple of 9.

(Reason)

8262=8000+200+60+2

=8*(999+1)+2*(99+1)+6*(9+1)+2

=(8*999+2*99+6*9)+(8+2+6+2)

What is firstly parenthesized is a multiple of 9, so you only have to check (8+2+6+2).

[Postscript]

Aside from the above, there are Quick Tables of Multiples for 7, 11 and 13 which are omitted here due to their poor practicability.

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