The operation of division is used in divisibility tests. A...

The operation of division is used in divisibility tests. A divisibility test allows us to determine whether a given number is divisible (without remainder) by another number. An integer is divisible by 8 if its last three digits form a numble by $8,$ and not otherwise. Show that (a) $2,923,296$ is divisible by 8 and (b) $7,291,623$ is not divisible by 8 .

The operation of division is used in divisibility tests. A divisibility test allows us to determine
whether a given number is divisible (without remainder) by another number.
An integer is divisible by 8 if its last three digits form a numble by $8,$ and not otherwise. Show that
(a) $2,923,296$ is divisible by 8 and
(b) $7,291,623$ is not divisible by 8 .

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Key Concepts

Divisibility

Divisibility refers to the property of one integer being able to be divided by another integer without leaving a remainder. It is a foundational concept in number theory and arithmetic, providing insight into factors and multiples of numbers.

Divisibility Tests

Divisibility tests are shortcuts or rules that allow one to determine whether one number is divisible by another without performing full division. These tests simplify the process by breaking down the problem into manageable steps, often involving specific parts of the number.

Rule for Divisibility by 8

The divisibility rule for 8 states that an integer is divisible by 8 if and only if its last three digits form a number that is a multiple of 8. This works because 1000 is divisible by 8, so any extra thousands do not affect whether the overall number is divisible by 8.

Remainder

The concept of a remainder arises in division when the dividend does not evenly divide by the divisor. If the remainder is zero after division, then the original number is exactly divisible by the divisor. This principle underlies all divisibility tests.

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