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This essential trains for: Math Kangaroo 3-4, Math Kangaroo 5-6, Math Kangaroo 7-8, SAT-I, GMAT, AMC-8, AMC-10.

For the study of divisibility it is useful to write the number in expanded form (or a variation thereof).

Take, for example the integer below and write it in an expanded form whereby you separate the last digit out:

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The first term, 1080 ends in zero, therefore it is automatically divisible by 2 and by 5 since:

equation

Therefore, if the other term - which is the last digit of the number - is divisible by 2 or by 5 the entire number is divisible by 2 or by 5, respectively.

Similarly, to study divisibility by 4, write the number in expanded form by separating the last two digits:

equation

The first term in the expansion is automatically divisible by 4 and by 25 since it is a multiple of 100:

equation

Therefore, if the other term - which is a number formed by the last two digits of the original number - is divisible by 4 or by 25 the entire number is divisible by 4 or by 25, respectively.

Similarly, to study divisibility by 8, write the number in expanded form by separating the last three digits:

equation

The first term in the expansion is automatically divisible by 8 and by 125 since it is a multiple of 1000:

equation

Therefore, if the other term - which is a number formed by the last three digits of the original number - is divisible by 8 or by 125 the entire number is divisible by 8 or by 125, respectively.

This reasoning reduces the complexity of establishing if a number is divisible by a power of 2 or of 5 and can be continued in a similar manner for any higher power of these two divisors.