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It’s important to note that the GMAT does not allow a calculator in the Quant section. Thus, it’s important to memorize and be able to use divisibility rules in answering certain questions. There are times when using GMAT divisibility rules to answer quant questions faster can help simplify numbers and prime factorization, and times when these rules can help determine whether a number is a multiple of another number.
Here Are the Topics We’ll Cover:
- GMAT Rules for Divisibility: An Overview
- Using Divisibility Rules for Prime Factorization Questions
- Using the Rules to Simplify Divisibility Questions
- Key Takeaways
- What’s Next?
GMAT Rules for Divisibility: An Overview
Before diving into specific examples, let’s review the divisibility rules for 0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.
Number Divisible by 0
No number is divisible by 0.
Number Divisible by 2
A number is divisible by 2 if the ones digit is 0, 2, 4, 6, or 8 – that is, if the units digit is even.
For example, 30, 42, 54, 66, and 78 are divisible by 2.
Number Divisible by 3
A number is divisible by 3 if the sum of all the digits is divisible by 3.
For example, 472,071 is divisible by 3 because the sum of its digits (4 + 7 + 2 + 0 + 7 + 1 = 21) is divisible by 3.
Number Divisible by 4
If the last two digits of a number are divisible by 4, then the number is divisible by 4.
For example, the last two digits of 244 are 44, which is divisible by 4. Students sometimes fail to see that a number that ends in 00 is divisible by 4. Just remember that all multiples of 100 are divisible by 4, since 100 = 25 x 4.
Number Divisible by 5
A number is divisible by 5 if the last (ones) digit is 0 or 5.
For example, the numbers 55 and 70 are divisible by 5.
Number Divisible by 6
A number is divisible by 6 if the number in question is an even number whose digits sum to a multiple of 3 (and therefore the number is divisible by both 2 and 3, the factors of 6).
For example, 522 is an even number, and its digits, 5, 2, and 2, sum to 9, a multiple of 3.
Number Divisible by 7
There are tricky formulas for this, but their logic is complicated. So, if you are asked whether a number is divisible by 7, just do the division.
Number Divisible by 8
If the number is even, divide the last three digits by 8. If there is no remainder, then the original number is divisible by 8.
For example, the number 1,160 is divisible by 8 because 160/8 = 20, which is an integer. Students often fail to see that if a number ends in 000, the number is divisible by 8. Just remember that all multiples of 1,000 are divisible by 8 because 1000 = 125 x 8.
Number Divisible by 9
A number is divisible by 9 if the sum of all the digits is divisible by 9.
For example, 479,655 is divisible by 9 because the sum of the digits (4 + 7 + 9 + 6 + 5 + 5 = 36) is divisible by 9.
Number Divisible by 10
If the ones digit is 0, then the number is divisible by 10.
For example, 10, 80, 90, 100, 1,120, and 10,000 are all divisible by 10.
Number Divisible by 11
A number is divisible by 11 if the sum of the odd-numbered place digits minus the sum of the even-numbered place digits is divisible by 11. The odd-numbered place digits are the 1st, 3rd, 5th, and so on digits to the left of the decimal point. Hence, they are the ones, hundreds, ten thousands, and so on digits. Similarly, the even-numbered place digits are the 2nd, 4th, 6th, and so on digits to the left of the decimal point. Hence, they are the tens, thousands, hundred-thousands, and so on digits.
For example, 253 is divisible by 11 because (2 + 3) – 5 = 0, which is divisible by 11 (remember, 0 is divisible by any number except itself). Likewise, 2,915 is divisible by 11 because (9 + 5) – (2 + 1) = 11, which is divisible by 11.
Number Divisible by 12
If a number is divisible by both 3 and 4, the number is also divisible by 12.
For example, since 24 is divisible by both 3 and 4, then 24 is also divisible by 12.
TTP PRO TIP:
Memorize the divisibility rules.
Now that we’ve reviewed the divisibility rules, let’s consider an example problem.
Example: Divisibility Rules
If x = 103, which of the following numbers could be added to x to create a sum that is divisible by 6?
- 0
- 1
- 2
- 4
- 5
Solution:
To be divisible by 6, a number must be divisible by 2 and 3.
For a number to be divisible by 2, it must be even, and for a number to be divisible by 3, the sum of the digits must be a multiple of 3.
Let’s go through the answer choices one by one.
Answer A: 103 + 0 = 103, which is not divisible by either 2 or 3 because it is not even and the digits do not add up to a number that is divisible by 3.
Answer B: 103 + 1 = 104, which is divisible by 2, but not by 3.
Answer C: 103 + 2 = 105, which is divisible by 3 but not by 2.
Answer D: 103 + 4 = 107, which is not divisible by either 2 or 3.
Answer E: 103 + 5 = 108, which is divisible by both 2 and 3.
Answer: E
In addition to knowing the divisibility rules to answer direct divisibility questions, knowing them helps when you encounter questions dealing with prime factorization.
Using Divisibility Rules for Prime Factorization Questions
Recall that prime factorization is the process of expressing positive numbers as the product of their prime factors. You should also recall that a prime number is one with only two factors: 1 and itself. Additionally, the first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.
Let’s say, for example, that we are given the number 288 and need to break 288 into prime factors. While we quickly see that 288 is an even number and thus divisible by 2, in order to most efficiently break 288 into prime factors, it is helpful to find a number larger than 2 (if applicable) that divides 288. Since the digits of 288 add up to 2 + 8 + 8 = 18, we see that 288 is divisible by 9, so we can begin the prime factorization by dividing 288 by 9.
288 = 9 * 32 = 3 * 3 * 8 * 4 = 3^2 * 2^3 * 2^2 = 3^2 * 2^5
Although what we did may seem insignificant, beginning your prime factorization by dividing by a large number allows you to take fewer steps when breaking numbers into prime factors, thus saving you time. If you are able to save even 10 seconds on every question, you will save 31 * 10 = 310 seconds, or almost 5 minutes, on the Quant section.
Consider the following example that combines using the divisibility rules and prime factorization.
Example: Prime Factorization and Divisibility
What prime factors do 234 nd 318 have in common?
I. 2
II. 3
III. 23
- I only
- II only
- I and II
- I, II, and III
- The two numbers share no prime factors in common.
Solution:
If your first step in prime factorizing 234 is dividing by 2 to get 2 x 117, you have a long road ahead. Instead, note that the sum of the digits of 234 is 2 + 3 + 4 = 9, which is divisible by 9. Thus, your first step in the prime factorization is:
234 = 9 x 26.
234 = 3 x 3 x 2 x 13
For the second number, 318, we note that since it is even, it is divisible by 2. But, we also note that the sum of the digits 3 + 1 + 8 = 12, which indicates that 318 is also divisible by 3. Thus, we use the divisibility rule for 6 to see that 318 is divisible by 6. Thus, we have:
318 = 6 x 53
318 = 2 x 3 x 53
Thus, we see that the numbers 234 and 318 share the two prime factors of 2 and 3.
Answer: C
Note that by taking a few seconds to use the divisibility rules rather than jumping in with division by 2, you have saved several steps in prime factorizing each of the numbers. Thus, you have saved as much as 30 seconds by not having to perform additional arithmetic steps.
TTP PRO TIP:
Use divisibility rules to streamline the prime factorization of large numbers.
Next, let’s discuss using divisibility rules when answering GMAT questions that directly involve divisibility.
Using the Rules to Simplify Divisibility Questions
We just discussed how we can use divisibility rules in questions that don’t directly ask about divisibility, but, of course, these rules also can help answer divisibility questions. Consider the following example:
Example: Simplifying a Problem by Using Divisibility Rules
If T = 213,425,212, then T + 2 must be divisible by which of the following?
I. 3
II. 6
III. 12
- I Only
- II Only
- I and II
- II and III
- I, II, and III
Solution:
T + 2 = 213,425,214. It would be time-prohibitive to divide each of the answer choices into T + 2. Instead, we’ll use the rules of divisibility. Because T + 2 ends in 4, which is an even number, it’s divisible by 2. In addition, the digits of T + 2 sum as follows: 2 + 1 + 3 + 4 + 2 + 5+ 2 + 1 + 4 = 24, which is a multiple of 3; thus, T + 2 is divisible by 3. So, we know that T + 2 is divisible by 2 and by 3. This also means that T + 2 is divisible by 6 because the prime factorization of 6 is 3 × 2. Thus, T + 2 is divisible by 3 and by 6.
Finally, we’ve learned that if a number is divisible by 3 and 4, it is also divisible by 12. Since we know that T + 2 is divisible by 3, we only need to determine whether it is also divisible by 4. We divide 14, the last two digits of T + 2, by 4. Since 14 is not divisible by 4, T + 2 is not divisible by 4, and thus is not divisible by 12.
Answer: C
If we don’t know our divisibility rules, the only way to solve this question is by dividing 213,425,212 + 2 = 213,425,214 by 3, 6, or 12. Doing so likely would be a long and error-prone process. On the other hand, using the divisibility rules for 3, 6, and 12, we can very easily come up with an answer.
KEY FACT:
Using the divisibility rules can make short work of an otherwise tough question.
Key Takeaways
As you can see, knowing divisibility rules like the back of your hand is a relatively simple way to significantly increase your efficiency and reduce your chances of making an error when answering not only GMAT math questions that deal directly with divisibility but also questions that deal with prime factorization.
What’s Next?
Remember, on the GMAT, you should seek to simplify numbers whenever possible; divisibility rules, which are part of our Number Properties chapter at our Target Test Prep course, are an important tool for doing just that. For more tips on how to efficiently answer GMAT Quant questions, check out our article with 10 strategies for earning a high GMAT Quant score without a calculator.
