Graduate Record Examinations (GRE) Quantitative Reasoning: Arithmetic and Number Properties Syllabus

Quantitative Reasoning: Arithmetic and Number Properties syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Quantitative Reasoning: Arithmetic and Number Properties in Graduate Record Examinations (GRE), not a summary of it.

1. 1

   Integers and Number Properties

   3 topics

   • Factors, multiples and divisibility
     • Prime factorisation
     • GCD and LCM
     • Divisibility rules
   • Even, odd, positive and negative properties
     • Operations on parity and sign
   • Remainders and modular thinking
     • Remainder patterns and cycles
2. 2

   Fractions, Decimals and Percentages

   3 topics

   • Fraction operations and comparison
     • Common denominators
     • Complex fractions
   • Decimal arithmetic and place value
     • Rounding and estimation
   • Percentage change and applications
     • Percent increase and decrease
     • Successive percentage changes
     • Simple and compound interest
3. 3

   Ratios, Proportion and Rates

   3 topics

   • Ratio setup and scaling
     • Part-to-part versus part-to-whole
   • Direct and inverse proportion
     • Constant of proportionality
   • Rate, work and speed problems
     • Combined work rates
     • Distance-speed-time
4. 4

   Powers, Roots and Estimation

   3 topics

   • Exponent rules
     • Negative and fractional exponents
     • Same-base manipulation
   • Roots and radicals
     • Simplifying surds
     • Rationalising
   • Scientific notation and approximation
     • Order of magnitude reasoning

Quantitative Reasoning: Arithmetic and Number Properties flashcards for Graduate Record Examinations (GRE)

24 of 51 cards from the Quantitative Reasoning: Arithmetic and Number Properties deck — real questions with worked answers.

1. What does it mean for an integer $a$ to be a factor (divisor) of an integer $b$?

   $a$ is a factor of $b$ if there exists an integer $k$ such that $b = a \cdot k$, i.e. $b$ divided by $a$ leaves no remainder ($a \mid b$).

2. How do you find the total number of factors of a positive integer from its prime factorization $n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$?

   Multiply one more than each exponent: number of factors $= (a_1 + 1)(a_2 + 1)\cdots(a_k + 1)$.

3. How many positive factors does $72$ have?

   Since $72 = 2^{3} \cdot 3^{2}$, the number of factors is $(3+1)(2+1) = 12$.

4. What is the relationship between the GCD and LCM of two positive integers $a$ and $b$?

   $\gcd(a,b) \times \operatorname{lcm}(a,b) = a \times b$.

5. State the divisibility rule for $3$ and for $9$.

   A number is divisible by $3$ if the sum of its digits is divisible by $3$; it is divisible by $9$ if the digit sum is divisible by $9$.

6. State the divisibility rule for $4$ and for $8$.

   Divisible by $4$ if the last two digits form a number divisible by $4$; divisible by $8$ if the last three digits form a number divisible by $8$.

7. State the divisibility rule for $11$.

   A number is divisible by $11$ if the alternating sum of its digits (e.g. $d_1 - d_2 + d_3 - \cdots$) is a multiple of $11$ (including $0$).

8. How is the Least Common Multiple (LCM) of several numbers found from prime factorizations?

   Take each prime that appears in any factorization, raised to the highest power it occurs in any number, then multiply these together.

9. How is the Greatest Common Divisor (GCD) found from prime factorizations?

   Take each prime common to all numbers, raised to the lowest power it appears, then multiply these together. Primes not shared are excluded.

10. What are the parity rules for addition: even $\pm$ even, odd $\pm$ odd, and even $\pm$ odd?

    even $\pm$ even $=$ even; odd $\pm$ odd $=$ even; even $\pm$ odd $=$ odd.

11. What are the parity rules for multiplication of integers?

    even $\times$ even $=$ even; even $\times$ odd $=$ even; odd $\times$ odd $=$ odd. A product is odd only if every factor is odd.

12. What sign results from multiplying or dividing two numbers with the same sign versus opposite signs?

    Same signs give a positive result; opposite signs give a negative result. ($(+)(+) = +$, $(-)(-) = +$, $(+)(-) = -$.)

13. Is $0$ considered even or odd, and what is its sign?

    $0$ is even (it is $2 \times 0$). It is neither positive nor negative.

14. When a negative number is raised to a power, what determines the sign of the result?

    A negative base raised to an even exponent is positive; raised to an odd exponent it is negative. E.g. $(-2)^{4} = 16$ but $(-2)^{3} = -8$.

15. In the division algorithm, how are dividend, divisor, quotient, and remainder related?

    $\text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder}$, where $0 \leq \text{remainder} < \text{divisor}$.

16. What is $17 \bmod 5$, and what does the result represent?

    $17 \bmod 5 = 2$, since $17 = 5 \times 3 + 2$; the result is the remainder when $17$ is divided by $5$.

17. If $a \equiv r_1 \pmod{n}$ and $b \equiv r_2 \pmod{n}$, what is $(a+b) \bmod n$ and $(a \cdot b) \bmod n$?

    $(a+b) \equiv (r_1 + r_2) \pmod{n}$ and $(a \cdot b) \equiv (r_1 \cdot r_2) \pmod{n}$. You can reduce each part to its remainder first.

18. How can you quickly find the units digit of a large power such as $7^{100}$ using cyclicity?

    The units digits of powers of $7$ cycle in a pattern of length $4$: $7, 9, 3, 1$. Compute $100 \bmod 4 = 0$, which corresponds to the last in the cycle, so the units digit is $1$.

19. What is the rule for adding two fractions $\frac{a}{b} + \frac{c}{d}$?

    $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$, then simplify. (Or use the least common denominator.)

20. How do you divide one fraction by another: $\frac{a}{b} \div \frac{c}{d}$?

    Multiply by the reciprocal of the divisor: $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{ad}{bc}$.

21. How do you compare two fractions $\frac{a}{b}$ and $\frac{c}{d}$ (with positive denominators) without finding a common denominator?

    Cross-multiply: $\dfrac{a}{b} > \dfrac{c}{d}$ if and only if $ad > bc$.

22. For positive fractions less than $1$, how does increasing the numerator versus the denominator affect the value?

    Increasing the numerator (denominator fixed) increases the fraction; increasing the denominator (numerator fixed) decreases the fraction.

23. What happens to a positive proper fraction's value when you add the same positive number to both numerator and denominator?

    A proper fraction ($<1$) moves closer to $1$, so it increases. (An improper fraction $>1$ would decrease toward $1$.)

24. How do you convert a fraction $\frac{a}{b}$ to a percentage?

    Compute $\dfrac{a}{b} \times 100\%$. For example, $\dfrac{3}{8} = 0.375 = 37.5\%$.

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