GMAT Online Course: Divisibility

Beyond the Basics: Mastering Divisibility Rules and Their Applications in GMAT Focus Edition Quantitative Skills.

In our previous discussions, we touched upon factors and multiples, inherently linked to the concept of divisibility. Now, we dedicate an entire lesson to divisibility itself, exploring it as a core property of numbers and dissecting advanced rules and applications crucial for GMAT success.

Understanding divisibility is not just about knowing if one number goes into another evenly; it's about discerning the structural properties of integers, which is vital for prime factorization, Least Common Multiple (LCM), Greatest Common Divisor (GCD), and solving complex number theory problems. This lesson will elevate your understanding from mere recognition to strategic application.

I. Defining Divisibility: The Core Concept Revisited

At its heart, divisibility is about exact division.

Formal Definition:

An integer A is said to be divisible by an integer B (where B is not zero) if, when A is divided by B, the remainder is 0.

Equivalently:

• A / B is an integer.

• A = B * k for some integer k.

When A is divisible by B, we also say that:

• B is a factor (or divisor) of A.

• A is a multiple of B.

Example:

• 40 is divisible by 8 because 40 / 8 = 5 (an integer, remainder 0).

• Therefore, 8 is a factor of 40, and 40 is a multiple of 8.

• 40 is NOT divisible by 7 because 40 / 7 = 5 with a remainder of 5.

II. Key Divisibility Rules: Beyond the Basics for Efficiency

While we covered basic rules earlier, let's consolidate and expand them, especially focusing on how they combine for composite numbers. These rules are indispensable for quickly analyzing numbers without resorting to long division.

• Divisibility by 2: Last digit is 0, 2, 4, 6, or 8 (i.e., the number is even).

  • Example: 7,492 (Yes, ends in 2).

• Divisibility by 3: The sum of the digits is divisible by 3.

  • Example: 1,572 (Sum = 1+5+7+2 = 15. 15 is divisible by 3, so 1,572 is).

• Divisibility by 4: The number formed by the last two digits is divisible by 4.

  • Example: 8,136 (36 is divisible by 4, so 8,136 is).

  • Tip: If the tens digit is odd, the last two digits must be 2 or 6 for divisibility by 4 (e.g., 12, 16, 32, 36, 52, 56, etc.). If the tens digit is even, the last two digits must be 0, 4, or 8 (e.g., 04, 08, 20, 24, etc.).

• Divisibility by 5: Last digit is 0 or 5.

  • Example: 9,305 (Yes, ends in 5).

• Divisibility by 6: The number is divisible by both 2 and 3. (Crucial: must pass both tests).

  • Example: 4,110 (Ends in 0, so divisible by 2. Sum = 4+1+1+0 = 6. 6 is divisible by 3, so 4,110 is. Since it passes both, it's divisible by 6).

• Divisibility by 8: The number formed by the last three digits is divisible by 8.

  • Example: 27,640 (640 is divisible by 8 (640 / 8 = 80), so 27,640 is).

• Divisibility by 9: The sum of the digits is divisible by 9.

  • Example: 3,465 (Sum = 3+4+6+5 = 18. 18 is divisible by 9, so 3,465 is).

• Divisibility by 10: Last digit is 0.

  • Example: 5,700 (Yes, ends in 0).

• Divisibility by 11: The alternating sum of the digits (starting from the rightmost digit, subtract the next, add the next, etc.) is divisible by 11.

  • Example: 1,364

    • 4 - 6 + 3 - 1 = 0. Since 0 is divisible by 11, 1,364 is divisible by 11.

  • Example: 9,878

    • 8 - 7 + 8 - 9 = 0. Since 0 is divisible by 11, 9,878 is divisible by 11.

• Divisibility by Composite Numbers (General Rule):

  If a number N is divisible by a composite number C, it must be divisible by all the prime factors of C. More precisely, if C = p1^a p2^b ..., then N must be divisible by p1^a, p2^b, etc.

  • Crucial Application: If C can be broken down into two coprime (or relatively prime - meaning their GCD is 1, they share no common prime factors) factors, then N must be divisible by both of those coprime factors.

    • Divisibility by 12: Check divisibility by 3 and 4 (since GCD(3,4)=1). Do NOT check by 2 and 6, because GCD(2,6)=2, meaning 2 is a shared factor and the rule breaks down.

      • Example: Is 384 divisible by 12?

        • By 3: 3+8+4 = 15 (divisible by 3). Yes.

        • By 4: 84 is divisible by 4 (84/4 = 21). Yes.

        • Since it's divisible by both 3 and 4, it's divisible by 12.

    • Divisibility by 15: Check divisibility by 3 and 5 (since GCD(3,5)=1).

    • Divisibility by 18: Check divisibility by 2 and 9 (since GCD(2,9)=1). Do NOT check by 3 and 6.

III. Divisibility in Expressions: How Operations Affect Divisibility

Understanding how divisibility behaves in sums, differences, and products is powerful for GMAT problems.

• Sums/Differences:

  • If A is divisible by K AND B is divisible by K, then (A + B) and (A - B) are also divisible by K.

    • Example: 20 is div by 4, 12 is div by 4. Then 20+12 = 32 (div by 4). 20-12 = 8 (div by 4).

  • If A is divisible by K but B is NOT divisible by K, then (A + B) and (A - B) are NOT divisible by K.

    • Example: 20 is div by 4, but 10 is NOT div by 4. Then 20+10 = 30 (NOT div by 4).

  • If NEITHER A NOR B is divisible by K, then (A + B) and (A - B) could be divisible by K or not. (Requires testing).

    • Example: 7 is NOT div by 3, 5 is NOT div by 3.

      • 7+5 = 12 (div by 3).

      • 7-5 = 2 (NOT div by 3).

• Products:

  • If A is divisible by K, then A * B is always divisible by K (for any integer B).

    • Example: 6 is div by 3. Then 6 5 = 30 (div by 3). 6 (-2) = -12 (div by 3).

  • If A * B is divisible by K, it does NOT necessarily mean A or B individually are divisible by K, unless K is a prime number.

    • Example: 12 is divisible by 6. 12 = 4 * 3. Neither 4 nor 3 are divisible by 6.

    • If K is prime: If A * B is divisible by a prime number P, then either A is divisible by P OR B is divisible by P (or both). This is a very important property for prime numbers.

IV. Connection to Remainders: The Zero Remainder Concept

Divisibility is fundamentally linked to remainders:

• If A is divisible by B, the remainder when A is divided by B is 0.

• If the remainder is not 0, then A is not divisible by B.

Example:

• 35 / 7 = 5 with remainder 0. So, 35 is divisible by 7.

• 37 / 7 = 5 with remainder 2. So, 37 is not divisible by 7.

V. Why Divisibility is Crucial for the GMAT

• Prime Factorization Foundation: Divisibility rules are your first line of defense when breaking down numbers into their prime factors.

• LCM/GCD: Directly applied in finding LCM and GCD, which are common GMAT topics.

• Number Properties: Questions about factors, multiples, number of factors, and properties of specific integers (e.g., "Is X a multiple of 15?") heavily rely on divisibility.

• Algebraic Expressions: Understanding divisibility rules for expressions helps in simplifying and analyzing algebraic terms.

• Data Sufficiency: Many DS questions test your knowledge of divisibility to determine sufficiency (e.g., "Is X divisible by 9?").

• Efficiency: Quickly testing divisibility saves significant time on complex calculations.

• Remainder Problems: A special case of remainder problems is when the remainder is 0, indicating divisibility.

VI. Common Pitfalls to Avoid

• Mixing Up Divisibility Rules: Confusing the rule for 4 with 8, or 3 with 9. Practice makes perfect.

• Incorrect Composite Divisor Breakdown: For a composite divisor C, ensure you break it into coprime factors for the rule to work (e.g., 12 needs 3 & 4, NOT 2 & 6).

• Assuming Integer Results: Remember that divisibility applies to integers dividing integers to produce an integer result. Don't apply rules to fractions or decimals.

• Missing Zero: Any non-zero integer divides 0. 0 is divisible by every non-zero integer.

• Negative Divisors: While mathematically valid, GMAT questions primarily focus on positive divisors unless explicitly stated.

Interactive Check Your Understanding:

1. Is 6,780 divisible by 4? By 6?

2. If an integer N is divisible by 10 and also by 3, is N necessarily divisible by 30? Why?

3. If A is not divisible by 5 and B is not divisible by 5, can A + B be divisible by 5? Give an example.

4. A number leaves a remainder of 0 when divided by 7. What does this tell us about the number's relationship with 7?

Practice Questions:

1. Which of the following numbers is divisible by both 9 and 4?

   a) 1,234

   b) 1,320

   c) 2,196

   d) 3,456

2. If x is an integer, and 3x + 12 is divisible by 3, which of the following must be true?

   a) x is divisible by 3.

   b) x is an even number.

   c) x + 4 is divisible by 3.

   d) x can be any integer.

3. How many integers between 100 and 200 (exclusive) are divisible by 11?

4. A company's total profit for the last year was $P. If $P is a 4-digit integer and is divisible by 2, 5, and 9, what is the smallest possible value of $P?

5. If A is divisible by 7 and B is divisible by 7, which of the following is NOT necessarily divisible by 7?

   a) A + B

   b) A - B

   c) A * B

   d) A / B (Assume B is a factor of A)

Solutions to Practice Questions:

1. Divisible by both 9 and 4:

   • For 9: Sum of digits must be divisible by 9.

   • For 4: Last two digits must be divisible by 4. a) 1,234: Sum=10 (No for 9). Last two digits 34 (No for 4). b) 1,320: Sum=6 (No for 9). Last two digits 20 (Yes for 4). c) 2,196: Sum=18 (Yes for 9). Last two digits 96 (Yes for 4). 96 / 4 = 24. d) 3,456: Sum=18 (Yes for 9). Last two digits 56 (Yes for 4). 56 / 4 = 14.

   Both c and d satisfy the conditions. In a GMAT setting, there would typically be only one correct answer unless specifically asked for "all that apply." If forced to choose, either 2,196 or 3,456 would be correct.

2. 3x + 12 is divisible by 3. What must be true about x?

   If 3x + 12 is divisible by 3, then 3(x + 4) is divisible by 3.

   Since 3(x+4) inherently has a factor of 3, it will always be divisible by 3, regardless of the value of x (as long as x is an integer).

   So, x can be any integer. For example, if x=1, 3(1)+12 = 15 (div by 3). If x=2, 3(2)+12 = 18 (div by 3). If x=0, 3(0)+12 = 12 (div by 3).

   The correct answer is d) x can be any integer.

3. Integers between 100 and 200 (exclusive) divisible by 11:

   This means 100 < 11k < 200.

   Find the first multiple of 11 greater than 100: 11 9 = 99, 11 10 = 110. So, k=10 is the first.

   Find the last multiple of 11 less than 200: 11 18 = 198, 11 19 = 209. So, k=18 is the last.

   The values of k are 10, 11, ..., 18.

   Number of integers = Last k - First k + 1 = 18 - 10 + 1 = 9.

4. Smallest 4-digit profit $P divisible by 2, 5, and 9:

   • Divisible by 2 and 5 implies divisible by 10 (since GCD(2,5)=1). So, P must end in 0.

   • Divisible by 10 and 9 implies divisible by 90 (since GCD(10,9)=1). We need the smallest 4-digit multiple of 90. Smallest 4-digit number is 1,000. 1000 / 90 = 11.11... So, the next integer multiple of 90 is 90 12. 90 12 = 1,080. The smallest possible value of $P is $1,080.

5. If A is divisible by 7 and B is divisible by 7, which is NOT necessarily divisible by 7?

   Let A = 7m and B = 7n for some integers m and n.

   a) A + B = 7m + 7n = 7(m + n). This is divisible by 7. (Necessarily)

   b) A - B = 7m - 7n = 7(m - n). This is divisible by 7. (Necessarily)

   c) A B = (7m) (7n) = 49mn = 7(7mn). This is divisible by 7. (Necessarily)

   d) A / B = (7m) / (7n) = m / n. This result is not necessarily an integer, let alone divisible by 7.

   * Example: A=14, B=7. Both are divisible by 7. A/B = 14/7 = 2. 2 is not divisible by 7.

   * Example: A=49, B=7. Both are divisible by 7. A/B = 49/7 = 7. 7 is divisible by 7.

   Since it's not necessarily divisible by 7 (and not even necessarily an integer), this is the correct answer.

   The correct answer is d) A / B.