GMAT Quant: Divisibility Rules and Shortcuts

Divisibility tests are valuable tools that can help you save time and make calculations easier on the GMAT Quant section. The most common tests focus on divisibility by 2, 3, 4, 5, 9, 10, 25, 50, and their higher powers. One of the most intriguing is the divisibility test for 11. Mastering these tests enables you to quickly spot factors and simplify problems without lengthy calculations. The following video and infographics will guide you through these tests with practical examples. Make sure to get comfortable with them, as they will be essential throughout your GMAT prep course.

 

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The Secret of Divisibility Rules

Sometimes you don’t need to look at a whole number to know if it’s divisible. You only need to look at the end of the number. 

The Divisibility Rule for 2, 5, and 10

To check if a number is divisible by 2, 5, or 10, you only look at the very last digit.

• For 2: If the last digit is even (0, 2, 4, 6, 8), it is divisible by 2.
• For 5: If the last digit is 0 or 5, it is divisible by 5.
• For 10: If the last digit is 0, it is divisible by 10.

Example: Take the number 87,6345.

The last digit is 5.

5 divided by 2 leaves a remainder of 1.

5 divided by 5 leaves a remainder of 0 (It is divisible!).

5 divided by 10 leaves a remainder of 5.

The Divisibility Rule for 4, 20, 25, 50, and 100

To check these numbers, you look at the last two digits.

• For 4: Check if the last two digits are divisible by 4.
• For 25: Check if the last two digits are 00, 25, 50, or 75.
• For 100: Check if the last two digits are 00.

Example: Take the number 98,879,63450.

The last two digits are 50.

50 divided by 4 leaves a remainder of 2.

50 divided by 25 leaves a remainder of 0 (It is divisible!).

50 divided by 100 leaves a remainder of 50.

The “Mother Rule” (Powers of 2 and 5)

There is a pattern here! The number of digits you check depends on the “power” of the number.

• To check 2^1 (which is 2) or 5^1 (which is 5): Look at the last 1 digit.
• To check 2^2 (which is 4) or 5^2 (which is 25): Look at the last 2 digits.
• To check 2^3 (which is 8) or 5^3 (which is 125): Look at the last 3 digits.
• To check 2^4 (which is 16) or 5^4 (which is 625): Look at the last 4 digits.

Example: Take the number 8,377,334,346,380.

To check for 8: Look at the last three digits (380).

380 divided by 8 leaves a remainder of 4.

To check for 125: Look at the last three digits (380).

380 divided by 125 leaves a remainder of 5.

Summary Table

 

The Divisibility Rules for 3 and 9: The Power of Addition

To check if a number is divisible by 3 or 9, you don’t look at the last digit. Instead, you add all the digits together.

• Step 1: Add every single digit in the number to get a “sum”.
• Step 2: Divide that sum by 3 or 9.
• Step 3: The remainder you get from that small division is the same as the remainder for the whole big number.

Example: 43457

• Sum of digits: 4 + 3 + 4 + 5 + 7 = 23.
• For 3: 23 divided by 3 leaves a remainder of 2.
• For 9: 23 divided by 9 leaves a remainder of 5.

The Divisibility Rules for 11: The Alternating Rule

Checking for 11 uses a unique “back and forth” addition method.

• Step 1: Add the digits in the odd positions (1st, 3rd, 5th, etc.).
• Step 2: Add the digits in the even positions (2nd, 4th, 6th, etc.).
• Step 3: Subtract the even sum from the odd sum to find the difference.
• Step 4: Divide that difference by 11 to find your remainder.
• Note: If the difference is negative, keep adding 11 until it becomes positive.

Example: 54321

• Odd digits: 1 + 3 + 5 = 9.
• Even digits: 2 + 4 = 6.
• Difference: 9 – 6 = 3.
• The remainder with 11 is 3.

Divisibility Tests by Combining Divisibility Rules: The “Broad Concept”

For many other numbers, you can simply combine the rules you already know. If a number is divisible by two smaller “partner” numbers, it is divisible by their product.

 

Suggestion: You don’t need to memorize an endless list of rules! Just learn the broad concepts for the core numbers and their higher powers; you can figure out the rest.

Divisibility Rules: Summarized Table

 

 

Key Takeaways:

• Divisibility by 2, 5, and 10: Simply check the last digit.
• Divisibility by 4, 25, and 100: Check the last two digits.
• Divisibility by higher powers (8, 125, 16, 625): Check the last three or four digits, depending on the power.
• Divisibility by 3, 9, and 11: Use special rules based on summing or alternating digits.
• Combining Rules: For numbers like 6, 12, 15, 18, and 22, use combinations of divisibility rules to simplify checking.

These rules, when mastered, can significantly speed up your calculations in the GMAT and other standardized tests. Use the concepts outlined here, and you’ll be ready to tackle any divisibility question quickly and efficiently!

 

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