In the following exercises, use the divisibility tests to determine whether each number is divis

step 1 So we want to determine if 78 is divisible by 2, 3, 4, 5, 6, and 10 using common divisibility te

In the following exercises, use the divisibility tests to...

Key Concepts

Divisibility by 10

A number is divisible by 10 if it ends in 0. This rule is straightforward due to the base-10 system, where any number with 0 in the units place is a multiple of 10.

Divisibility by 5

A number is divisible by 5 if its last digit is either 0 or 5. This rule reflects the structure of the decimal system, as any number ending in these digits aligns with the multiples of 5.

Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. This rule is based on the fact that 100 is divisible by 4, so every additional hundred follows suit, making the last two digits decisive.

Divisibility by 6

A number is divisible by 6 if it meets the divisibility criteria for both 2 and 3. This rule is based on the prime factors of 6, and by confirming divisibility by 2 and 3 separately, one can conclude divisibility by 6.

Divisibility by 2

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). This rule takes advantage of the fact that in the base-10 system, the parity of the entire number is determined solely by its units digit.

Divisibility Rules

Divisibility rules are shortcuts that allow one to determine if a number is divisible by another without performing full division. They are based on specific properties of numbers in our decimal system and help simplify the process of identifying factors by analyzing key features like digit sums or last digits.

Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. This rule works because each power of 10 is congruent to 1 modulo 3, allowing the overall sum of the digits to reflect the number's divisibility by 3.

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