Divisibility rules
48 flashcards covering Divisibility rules for the GMAT Quantitative section.
Divisibility rules are straightforward methods to quickly check if one number divides another evenly, without any remainder. For example, a number is divisible by 2 if it's even, by 3 if the sum of its digits is divisible by 3, and by 5 if it ends in 0 or 5. These rules simplify working with factors, multiples, and fractions, saving time in everyday math and problem-solving.
On the GMAT Quantitative section, divisibility rules appear in number properties questions, such as finding greatest common divisors or simplifying expressions in data sufficiency and problem-solving formats. Common traps include misapplying rules for numbers like 4 or 9, or overlooking edge cases with larger numbers, which can lead to incorrect answers. Focus on memorizing and practicing rules for key divisors like 2, 3, 5, 9, and 10 to handle these efficiently and boost your accuracy.
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Terms (48)
- 01
Definition of divisibility
Divisibility means that one number can be divided by another without leaving a remainder, so if A is divisible by B, then A divided by B results in a whole number.
- 02
Divisibility rule for 2
A number is divisible by 2 if its last digit is even, meaning it ends in 0, 2, 4, 6, or 8.
- 03
Divisibility rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
- 04
Divisibility rule for 4
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- 05
Divisibility rule for 5
A number is divisible by 5 if its last digit is 0 or 5.
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Divisibility rule for 6
A number is divisible by 6 if it is divisible by both 2 and 3.
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Divisibility rule for 7
To check if a number is divisible by 7, double the last digit and subtract it from the rest of the number; if the result is divisible by 7, so is the original number.
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Divisibility rule for 8
A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
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Divisibility rule for 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
- 10
Divisibility rule for 10
A number is divisible by 10 if its last digit is 0.
- 11
Divisibility rule for 11
A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is a multiple of 11, including 0.
- 12
Divisibility rule for 12
A number is divisible by 12 if it is divisible by both 3 and 4.
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Divisibility rule for 13
To check if a number is divisible by 13, multiply the last digit by 4, add it to the rest of the number, and see if the result is divisible by 13.
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Divisibility rule for 15
A number is divisible by 15 if it is divisible by both 3 and 5.
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Divisibility rule for 25
A number is divisible by 25 if its last two digits are 00, 25, 50, or 75.
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Common trap with rule for 2
Some might think a number is divisible by 2 only if it ends in an even digit other than 0, but 0 is also valid, as in 10.
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Common trap with rule for 3
Students often forget to check if the sum of digits is exactly divisible by 3, not just close, like mistaking 12 (sum 3) for not divisible when it is.
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Common trap with rule for 4
Confusing it with the rule for 2 by only checking the last digit, ignoring that the last two digits must form a number divisible by 4.
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Common trap with rule for 5
Assuming a number ending in 5 is always divisible by 5, which is true, but forgetting that ending in 0 also works.
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Common trap with rule for 6
Overlooking that a number must satisfy both the rules for 2 and 3, not just one, like thinking 9 is divisible by 6 because it's divisible by 3.
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Common trap with rule for 7
The subtraction method can lead to errors if not applied carefully, such as miscalculating the doubled last digit.
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Common trap with rule for 8
Mistaking it for the rule for 4 by only checking the last two digits, when actually the last three must be divisible by 8.
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Common trap with rule for 9
Similar to the rule for 3, students might think a sum of digits like 18 is not divisible by 9 because 18 divided by 9 is 2, which it is.
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Common trap with rule for 10
Believing that only numbers ending in 00 are divisible by 10, ignoring single zeros like 20.
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Common trap with rule for 11
Forgetting that the alternating sum could be negative or zero and still count as a multiple of 11.
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Strategy for checking multiple divisibility
When a problem requires checking divisibility by a composite number, break it down into its prime factors and apply the rules for those factors.
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Using sum of digits for 3 and 9
For both 3 and 9, repeatedly sum the digits until you get a single digit; if that digit is 3, 6, or 9 for 3, or exactly 9 for 9, the number is divisible.
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Difference between rules for 3 and 9
While both use the sum of digits, for 3 the sum must be divisible by 3, but for 9 it must be divisible by 9, making 9 more restrictive.
- 29
Applying divisibility for prime numbers
For prime numbers like 7 or 11, use their specific rules since they don't factor into smaller integers beyond 1 and themselves.
- 30
Example of rule for 2
To determine if 246 is divisible by 2, check the last digit: it is 6, which is even, so yes.
divided by 2 equals 123.
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Example of rule for 3
For 123, sum the digits: 1 + 2 + 3 = 6, and 6 is divisible by 3, so 123 is divisible by 3.
divided by 3 equals 41.
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Example of rule for 4
For 124, look at the last two digits: 24, and 24 divided by 4 is 6, so 124 is divisible by 4.
divided by 4 equals 31.
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Example of rule for 5
For 125, the last digit is 5, so it is divisible by 5.
divided by 5 equals 25.
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Example of rule for 6
For 126, it ends in 6 (divisible by 2) and sum of digits is 1 + 2 + 6 = 9 (divisible by 3), so it is divisible by 6.
divided by 6 equals 21.
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Example of rule for 7
For 343, double the last digit 3 to get 6, subtract from 34: 34 - 6 = 28, and 28 is divisible by 7, so 343 is.
divided by 7 equals 49.
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Example of rule for 8
For 104, the last three digits are 104, and 104 divided by 8 is 13, so it is divisible by 8.
divided by 8 equals 13.
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Example of rule for 9
For 81, sum of digits is 8 + 1 = 9, which is divisible by 9, so 81 is.
divided by 9 equals 9.
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Example of rule for 10
For 150, the last digit is 0, so it is divisible by 10.
divided by 10 equals 15.
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Example of rule for 11
For 121, alternating sum is 1 - 2 + 1 = 0, which is a multiple of 11, so 121 is divisible by 11.
divided by 11 equals 11.
- 40
Example of rule for 12
For 144, it is divisible by 3 (sum 1+4+4=9) and by 4 (last two digits 44 divisible by 4), so by 12.
divided by 12 equals 12.
- 41
Nuance in rule for large numbers and 3
For very large numbers, sum the digits multiple times if needed to simplify checking divisibility by 3.
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Nuance in rule for 4 and trailing zeros
Numbers with trailing zeros are often divisible by 4 if the zeros make the last two digits even and appropriately formed.
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When to use divisibility rules in GMAT problems
Divisibility rules help quickly eliminate options or verify factors in problems involving remainders, factors, or multiples.
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Divisibility by powers of 2
For powers like 4 (2 squared) or 8 (2 cubed), check the last two or three digits respectively.
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Divisibility by powers of 5
For 25 (5 squared), check the last two digits as specified earlier.
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Combining rules for 18
A number is divisible by 18 if it is divisible by both 2 and 9.
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Edge case for rule for 11 with single-digit numbers
Single-digit numbers are not divisible by 11 except for 0, but 11 itself is the smallest positive example.
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Why divisibility rules work
These rules are based on modular arithmetic, where properties of numbers modulo the divisor remain consistent.