Remainders
49 flashcards covering Remainders for the GMAT Quantitative section.
Remainders are the amounts left over after dividing one number by another. For example, when you divide 17 by 5, you get 3 as the quotient and 2 as the remainder, since 5 goes into 17 three times with 2 left over. This concept is a key part of basic arithmetic and helps in understanding how numbers relate through division, especially in problems involving patterns or cycles.
On the GMAT Quantitative section, remainders appear in questions testing number properties, often within data sufficiency or problem-solving formats. You'll encounter scenarios like finding remainders in modular arithmetic, divisibility rules, or word problems with real-world contexts. Common traps include overlooking negative numbers or confusing remainders with quotients, so focus on practicing the modulo operation and double-checking calculations. Mastering this topic can save time on the test by simplifying complex division problems.
Remember to always verify your remainder by performing a quick division check.
Terms (49)
- 01
Definition of Remainder
The remainder is the amount left over after dividing one integer by another, always less than the divisor and non-negative in standard contexts.
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Dividend in Division
In division, the dividend is the number being divided, from which the quotient and remainder are derived when dividing by another integer.
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Divisor in Division
The divisor is the number by which another integer is divided, determining how many times it fits evenly and what remainder is left.
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Quotient in Division
The quotient is the integer result of dividing one number by another, representing how many times the divisor fits into the dividend before the remainder.
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How to Find a Remainder
To find a remainder, divide the dividend by the divisor using long division or a calculator, then subtract the product of the quotient and divisor from the dividend.
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Remainder When Dividing by 2
When dividing an integer by 2, the remainder is 0 if the number is even and 1 if it is odd, helping to classify numbers as even or odd.
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Remainder When Dividing by 3
When dividing by 3, the remainder can be 0, 1, or 2, which is useful for checking divisibility or patterns in sequences.
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Remainder When Dividing by 5
When dividing by 5, the remainder is a number from 0 to 4, often used to determine the last digit's properties in base 10.
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Remainder When Dividing by 10
When dividing by 10, the remainder is the last digit of the number, as it represents the units place value.
- 10
Modular Arithmetic Basics
Modular arithmetic deals with remainders, where two numbers are congruent modulo m if they have the same remainder when divided by m.
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Congruence Modulo m
Two integers a and b are congruent modulo m if m divides (a - b) exactly, meaning they leave the same remainder when divided by m.
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Adding Remainders
To add two numbers and find the remainder when divided by m, add the numbers first, then take the remainder of the sum divided by m.
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Subtracting Remainders
To subtract two numbers and find the remainder when divided by m, subtract them first, then adjust to ensure the result is between 0 and m-1.
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Multiplying Remainders
To multiply two numbers and find the remainder when divided by m, multiply them first, then take the remainder of the product divided by m.
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Remainder of a Product
The remainder of a product a b divided by m is the same as the product of the remainders of a and b divided by m.
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Remainder of a Sum
The remainder of a sum a + b divided by m is the same as the sum of the remainders of a and b divided by m.
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Negative Numbers and Remainders
For negative dividends, the remainder is typically adjusted to be non-negative, such as adding the divisor to make it positive before finding the remainder.
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Remainder with Exponents
To find the remainder of a number raised to a power divided by m, compute the power modulo m, which simplifies large exponents.
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Cyclicity in Remainders
Cyclicity refers to the repeating pattern of remainders when powers of a number are divided by m, such as the cycle of last digits for powers of 2.
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Last Digit of a Number
The last digit of a number is its remainder when divided by 10, which is crucial for problems involving units place or trailing digits.
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Binomial Theorem and Remainders
In the binomial theorem, remainders can help find the last few digits of expansions by computing modulo 10 or another small number.
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Euclidean Algorithm
The Euclidean algorithm uses repeated division and remainders to find the greatest common divisor of two numbers efficiently.
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Remainder in Equations
In equations involving remainders, solve for variables by expressing conditions as congruences, such as x ≡ r mod m.
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Divisibility Rules and Remainders
Divisibility rules check if a number has a remainder of 0 when divided by another, like a number is divisible by 3 if the sum of its digits is.
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Common Trap: Ignoring Negative Remainders
A common error is not adjusting negative results to positive remainders, which can lead to incorrect answers in modular problems.
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Strategy for Large Exponents
For large exponents in remainder problems, use patterns or modular exponentiation to reduce calculations, like finding cycles modulo m.
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Worked Example: 17 Divided by 5
When 17 is divided by 5, the quotient is 3 and the remainder is 2, since 5 times 3 is 15, and 17 minus 15 equals 2.
÷ 5 = 3 remainder 2
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Worked Example: 2^3 Modulo 7
raised to the power of 3 is 8, and 8 divided by 7 has a remainder of 1, so 2^3 ≡ 1 mod 7.
^3 = 8, 8 - 7 = 1
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Worked Example: Sum of Remainders
If a leaves remainder 3 when divided by 5 and b leaves remainder 4, their sum leaves remainder 2 when divided by 5, since 3 + 4 = 7, and 7 mod 5 = 2.
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Worked Example: Product Remainder
If a is 4 mod 6 and b is 5 mod 6, their product is 20 mod 6, which is 2, since 20 divided by 6 leaves a remainder of 2.
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Remainder in Word Problems
In word problems, remainders often represent leftovers, like items not fitting evenly into groups, requiring setup of equations with congruences.
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Properties of Zero Remainder
A remainder of zero means the dividend is exactly divisible by the divisor, indicating the dividend is a multiple of the divisor.
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Remainder and Fractions
Remainders relate to fractions as the numerator part not covered by the whole divisions, though GMAT focuses on integer remainders.
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Multiple Divisors and Remainders
When a number has specified remainders for different divisors, solve using simultaneous congruences for the least common multiple.
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Pattern Recognition in Remainders
Identifying patterns, like every third number in a sequence having the same remainder, helps solve problems without full calculations.
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Remainder with Decimals
Though GMAT emphasizes integers, remainders can conceptually apply to decimals by considering the fractional part, but focus on integer contexts.
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Trap: Confusing Quotient and Remainder
Mixing up the quotient and remainder can lead to errors, as the quotient is the integer part of division, while the remainder is the leftover.
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Strategy: Use Substitution for Remainders
Replace a number with its remainder equivalent to simplify equations, such as letting x = mk + r to solve problems.
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Worked Example: 123 Divided by 7
divided by 7 gives a quotient of 17 and a remainder of 4, since 7 times 17 is 119, and 123 minus 119 is 4.
÷ 7 = 17 remainder 4
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Worked Example: 3^4 Modulo 5
raised to the power of 4 is 81, and 81 divided by 5 has a remainder of 1, so 3^4 ≡ 1 mod 5.
^4 = 81, 81 - 165 = 1
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Remainder in Sequences
In arithmetic sequences, remainders can help identify terms that meet specific divisibility conditions.
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Inverse in Modular Arithmetic
The modular inverse of a number a modulo m is a number b such that ab ≡ 1 mod m, used in solving linear congruences.
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Linear Congruences
A linear congruence is an equation like ax ≡ b mod m, solved by finding x that satisfies the remainder condition.
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Common Factor in Remainders
If two numbers share a common factor with the modulus, it affects whether congruences have solutions.
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Trap: Overlooking Zero Divisor
Dividing by zero is undefined, so ensure the divisor in remainder problems is a positive integer greater than zero.
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Strategy: Break Down Large Numbers
For large dividends, break them into smaller parts or use properties to find remainders without full division.
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Worked Example: Remainder of 100! Divided by 7
To find the remainder of 100! divided by 7, note that 100! includes multiples of 7, so it is divisible by 7, leaving a remainder of 0.
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Remainder and Prime Numbers
Prime numbers as divisors simplify remainder problems since they have no factors other than 1 and themselves.
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Composite Divisors and Remainders
With composite divisors, remainders can be analyzed by factoring the divisor into primes.