GMAT Online Course: Quotients and Remainders

The Art of the Leftover: Mastering Remainders for GMAT Success.

While divisibility tells us when a number divides another perfectly, real-world division often leaves something behind: a remainder. Understanding remainders goes far beyond simple arithmetic; it's a critical concept in GMAT quantitative reasoning, forming the basis for advanced number properties problems, cyclicity, and even certain data sufficiency scenarios. This lesson will provide a comprehensive look at what remainders are, how they behave under various operations, and the powerful techniques for manipulating them to solve complex problems efficiently.

I. Defining Remainders: The "Leftovers" of Division

When one integer is divided by another, and the division is not exact, the amount left over is called the remainder.

The Division Algorithm (Euclidean Algorithm):

For any integer Dividend (D) and a positive integer Divisor (d), there exist unique integers Quotient (q) and Remainder (r) such that:

D = d * q + r

where 0 ≤ r < d.

• Dividend (D): The number being divided.

• Divisor (d): The number by which you are dividing.

• Quotient (q): The whole number of times the divisor goes into the dividend.

• Remainder (r): The amount left over.

Key Rule: The remainder r must always be non-negative (zero or positive) and strictly less than the divisor d.

Examples:

• 17 ÷ 5:

  • 17 = 5 * 3 + 2

  • Dividend = 17, Divisor = 5, Quotient = 3, Remainder = 2. (0 ≤ 2 < 5 is true).

• 24 ÷ 6:

  • 24 = 6 * 4 + 0

  • Remainder = 0. This indicates that 24 is divisible by 6.

• -13 ÷ 4: (This requires careful thought, as GMAT prefers positive remainders)

  • If we think 4 (-3) = -12, then -13 = 4 (-3) - 1. Remainder is -1.

  • To get a positive remainder: We need 4 * q to be less than or equal to -13 but as close as possible.

    • 4 (-4) = -16. So, -13 = 4 (-4) + 3.

    • Quotient = -4, Remainder = 3. (0 ≤ 3 < 4 is true).

  • GMAT Rule: Unless otherwise stated, assume the remainder is always positive.

II. Properties of Remainders: How They Behave in Operations

These properties are incredibly powerful for solving remainder problems, especially with large numbers or expressions. The notation A mod d means "the remainder when A is divided by d."

1. Remainder of a Sum or Difference:

To find the remainder of a sum or difference, you can find the remainder of each term first, then sum/subtract those remainders, and finally find the remainder of that result.

• (A + B) mod d = (A mod d+B mod d) mod d

• (A - B) mod d = (A mod d-B mod d) mod d

  Example: What is the remainder when (47 + 73) is divided by 5?

  • Method 1 (Direct): 47 + 73 = 120. 120 mod 5 = 0.

  • Method 2 (Using Property):

    • 47 mod 5 = 2

    • 73 mod 5 = 3

    • (2 + 3) mod 5 = 5 mod 5 = 0. (Matches!)

  Example: What is the remainder when (85 - 12) is divided by 7?

  • Method 1 (Direct): 85 - 12 = 73. 73 mod 7 = 3.

  • Method 2 (Using Property):

    • 85 mod 7 = 1 (85 = 7*12 + 1)

    • 12 mod 7 = 5 (12 = 7*1 + 5)

    • (1 - 5) mod 7 = (-4) mod 7. To get a positive remainder, add the divisor: -4 + 7 = 3. (Matches!)

2. Remainder of a Product:

To find the remainder of a product, you can find the remainder of each factor first, then multiply those remainders, and finally find the remainder of that result.

• (A B) mod d = (A mod dB mod d) mod d

  Example: What is the remainder when (23 * 17) is divided by 5?

  • Method 1 (Direct): 23 * 17 = 391. 391 mod 5 = 1 (since 391 ends in 1).

  • Method 2 (Using Property):

    • 23 mod 5 = 3

    • 17 mod 5 = 2

    • (3 * 2) mod 5 = 6 mod 5 = 1. (Matches!)

3. Remainder of a Power (Cyclicity):

Finding the remainder of a large power A^x mod d often involves identifying a repeating pattern (cyclicity) in the remainders of successive powers of the base A.

• Example: What is the remainder when 3^100 is divided by 7?

  • 3^1 mod 7 = 3

  • 3^2 mod 7 = 9 mod 7 = 2

  • 3^3 mod 7 = (3^2 3^1) mod 7 = (2 3) mod 7 = 6

  • 3^4 mod 7 = (3^3 3^1) mod 7 = (6 3) mod 7 = 18 mod 7 = 4

  • 3^5 mod 7 = (3^4 3^1) mod 7 = (4 3) mod 7 = 12 mod 7 = 5

  • 3^6 mod 7 = (3^5 3^1) mod 7 = (5 3) mod 7 = 15 mod 7 = 1

  • The remainder sequence is 3, 2, 6, 4, 5, 1, and it repeats every 6 powers (the cycle length is 6).

  • To find 3^100 mod 7, find the remainder of the exponent (100) when divided by the cycle length (6).

    • 100 mod 6 = 4 (since 100 = 6 * 16 + 4).

  • This means 3^100 will have the same remainder as 3^4 when divided by 7.

  • 3^4 mod 7 = 4.

  • So, the remainder when 3^100 is divided by 7 is 4.

III. Common Applications and Variations on the GMAT

• Finding a missing digit: Using remainder rules (especially divisibility rules which are remainders of 0).

• Remainder in a sequence/pattern: Identifying the remainder of a specific term based on a repeating pattern.

• Properties of numbers: Questions like "If N leaves a remainder of X when divided by Y..."

• "Least possible value" or "Smallest integer" problems: Often involve remainders and constructing numbers that satisfy certain remainder conditions.

• Negative Remainders (Implied): While GMAT expects positive remainders, understanding that a mod d = r is equivalent to a = qd + r and a = (q+1)d + (r-d) can be helpful.

  • A remainder of -2 when divided by 5 is the same as 5 - 2 = 3.

  • A number N that is 2 less than a multiple of 5 (N = 5k - 2) has a remainder of 3 when divided by 5 (N = 5(k-1) + 3).

IV. Why Remainders are Crucial for the GMAT

• Direct Question Type: Remainder problems appear frequently, testing your understanding of the definition and properties.

• Foundation for Cyclicity: Many questions involving units digits, powers, or repeating patterns rely on remainder cyclicity.

• Number Properties Integration: Remainder concepts are often combined with factors, multiples, even/odd, and prime numbers.

• Problem-Solving Efficiency: Applying remainder properties can drastically simplify calculations that would otherwise be long and prone to error.

• Data Sufficiency: Determining the remainder of an expression is a common task in DS.

V. Common Pitfalls to Avoid

• Incorrect Remainder Range: Always ensure 0 ≤ r < d. Forgetting the "less than divisor" part is critical.

• Negative Remainders: While useful conceptually, convert to positive for GMAT answers. A remainder of -1 when divided by 4 means a remainder of 3.

• Direct Division for Large Numbers: Don't try to directly divide 3^100 by 7; use properties or cyclicity.

• Misapplying Properties: Be careful with the order of operations for sum/product/power properties.

• Ignoring the Divisor: The divisor d is paramount. All remainder calculations are relative to d.

Check Your Understanding:

1. What is the remainder when 98 is divided by 9?

2. What is the remainder when (123 + 456) is divided by 10?

3. What is the units digit of 7^23? (Hint: The units digit is the remainder when divided by 10).

4. If a number N leaves a remainder of 3 when divided by 7, what is the remainder when 2N is divided by 7?

Practice Questions:

1. When a positive integer X is divided by 6, the remainder is 4. What is the remainder when 3X is divided by 6?

   a) 0

   b) 1

   c) 2

   d) 4

2. If N is an integer, what is the remainder when (N + 2)(N + 3) is divided by 5 if N leaves a remainder of 1 when divided by 5?

3. What is the remainder when 123,456,789 is divided by 9?

4. A certain number of cookies were distributed among 8 children. Each child received an equal number of cookies, and 5 cookies were left over. If the same number of cookies were distributed among 6 children, what is the smallest possible number of cookies that would be left over?

5. What is the units digit of 2^105?

Solutions to Practice Questions:

1. Remainder of 3X when X divided by 6 is 4:

   Given X mod 6 = 4. This means X = 6k + 4 for some integer k.

   We want 3X mod 6.

   3X = 3(6k + 4) = 18k + 12.

   18k is clearly divisible by 6, so 18k mod 6 = 0.

   12 is also divisible by 6, so 12 mod 6 = 0.

   Therefore, (18k + 12) mod 6 = (0 + 0) mod 6 = 0.

   The correct answer is a) 0.

2. Remainder of (N + 2)(N + 3) when divided by 5, if N mod 5 = 1:

   Given N mod 5 = 1.

   Then, (N + 2) mod 5 = (1 + 2) mod 5 = 3 mod 5 = 3.

   And (N + 3) mod 5 = (1 + 3) mod 5 = 4 mod 5 = 4.

   Now, apply the product property:

   (N + 2)(N + 3) mod 5 = ( (N+2) mod 5 * (N+3) mod 5 ) mod 5

   = (3 * 4) mod 5

   = 12 mod 5

   = 2.

3. Remainder of 123,456,789 divided by 9:

   A number is divisible by 9 if the sum of its digits is divisible by 9. The remainder when divided by 9 is the same as the remainder when the sum of its digits is divided by 9.

   Sum of digits = 1+2+3+4+5+6+7+8+9 = 45.

   45 mod 9 = 0 (since 45 is divisible by 9).

   The remainder is 0.

4. Cookies problem (remainder from 8 is 5, smallest remainder from 6):

   Let the number of cookies be C.

   C mod 8 = 5 implies C = 8k + 5 for some integer k.

   Possible values for C: 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, ...

   We need to find the remainder when these numbers are divided by 6.

   • 5 mod 6 = 5

   • 13 mod 6 = 1

   • 21 mod 6 = 3

   • 29 mod 6 = 5

   • 37 mod 6 = 1 The remainders cycle 5, 1, 3, 5, 1, 3... The smallest possible number of cookies left over (smallest remainder) is 1.

5. Units digit of 2^105:

   The units digit of a power is the remainder when that power is divided by 10. We need to find the cyclicity of the units digits of powers of 2.

   • 2^1 = 2

   • 2^2 = 4

   • 2^3 = 8

   • 2^4 = 16 -> units digit is 6

   • 2^5 = 32 -> units digit is 2 The cycle of units digits is 2, 4, 8, 6. The cycle length is 4. To find the units digit of 2^105, find the remainder of the exponent (105) when divided by the cycle length (4). 105 mod 4 = 1 (since 105 = 4 * 26 + 1). This means the units digit of 2^105 will be the same as the 1st digit in the cycle, which is 2.