Divisibility rules are a set of guidelines that help determine whether a given integer is divisible by another integer without leaving a remainder. These rules provide a systematic way to quickly assess the divisibility of numbers, which is an important concept in elementary algebra and number theory.
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Divisibility rules are particularly useful for determining the divisibility of large numbers without the need for long division.
The divisibility rules for the numbers 2, 3, 4, 5, 6, 8, 9, and 10 are commonly tested in elementary algebra.
A number is divisible by 2 if its last digit is divisible by 2 (i.e., the number is even).
A number is divisible by 3 if the sum of its digits is divisible by 3.
A number is divisible by 4 if its last two digits are divisible by 4.
Review Questions
Explain how the divisibility rule for 2 works and provide an example.
The divisibility rule for 2 states that a number is divisible by 2 if its last digit is divisible by 2, meaning the number is even. For example, the number 246 is divisible by 2 because the last digit, 6, is even. Similarly, the number 357 is not divisible by 2 because the last digit, 7, is odd.
Describe the divisibility rule for 3 and explain how it differs from the rule for 2.
The divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3. This rule is different from the divisibility rule for 2 because it looks at the sum of the digits, rather than just the last digit. For instance, the number 123 is divisible by 3 because the sum of its digits (1 + 2 + 3 = 6) is divisible by 3, whereas the number 246 is not divisible by 3 because the sum of its digits (2 + 4 + 6 = 12) is not divisible by 3.
Analyze the divisibility rule for 4 and explain how it relates to the concept of place value.
The divisibility rule for 4 states that a number is divisible by 4 if its last two digits are divisible by 4. This rule is based on the concept of place value, where the last two digits of a number represent the ones and tens places. By checking the divisibility of the last two digits, you can determine the divisibility of the entire number by 4. For example, the number 1,236 is divisible by 4 because the last two digits, 36, are divisible by 4, whereas the number 1,237 is not divisible by 4 because the last two digits, 37, are not divisible by 4.
Related terms
Divisor: A divisor is a number that divides another number without leaving a remainder.
Dividend: A dividend is a number that is being divided by another number.
Remainder: The remainder is the amount left over when one number is divided by another number.