6.1 Number Properties, Divisibility Rules & Prime Factorization

Number Properties, Divisibility Rules & Prime Factorization

Number theory forms the structural foundation of the Quantitative Reasoning section on the National Testing Service (NTS) GAT General examination. A significant portion of quantitative questions evaluates your mastery of integer properties, divisibility shortcuts, prime factor decomposition, and modular patterns. By mastering these fundamental arithmetic principles, you can bypass lengthier long division and algebraic operations, solving GAT quantitative items with speed and precision.

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1. Classification of Real Numbers & Arithmetic Parity Rules

Understanding the real number system and the behavioral properties of odd and even integers is essential for solving quantitative comparison and algebraic sufficiency problems on the GAT.

Classification Breakdown

• Real Numbers ($\mathbb{R}$): All numbers on the continuous number line, divided into rational and irrational numbers.
• Rational Numbers ($\mathbb{Q}$): Numbers expressible in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Examples include $\frac{3}{4}$, $-7$, $0.25$, and $0.\overline{33}$.
• Irrational Numbers: Non-repeating, non-terminating decimals that cannot be written as fractions. Examples include $\sqrt{2}$, $\sqrt{5}$, $\pi$, and $e$.
• Integers ($\mathbb{Z}$): Complete positive and negative whole numbers including zero: $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
• Whole Numbers: Non-negative integers: $\{0, 1, 2, 3, \dots\}$.
• Natural Numbers ($\mathbb{N}$): Positive counting integers: $\{1, 2, 3, \dots\}$.
• Prime Numbers: Positive integers strictly greater than $1$ that have exactly two positive divisors: $1$ and the number itself. The first ten primes are $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
  • Note: $2$ is the only even prime number and the smallest prime number. $1$ is neither prime nor composite.
• Composite Numbers: Positive integers greater than $1$ that have more than two positive divisors (e.g., $4, 6, 8, 9, 10$).

Parity Rules for Even and Odd Integers

Let $E$ represent an even integer ($\{0, \pm 2, \pm 4, \dots\}$) and $O$ represent an odd integer ($\{\pm 1, \pm 3, \pm 5, \dots\}$):

Key Exam Insight: If the product of a set of integers is odd, every single integer in that set must be odd. If the product of a set of integers is even, at least one integer in the set must be even.

Properties of Consecutive Integers

• The sum of $n$ consecutive integers is divisible by $n$ if $n$ is an odd integer (e.g., $3 + 4 + 5 = 12$, which is divisible by $3$).
• The product of $k$ consecutive integers is always divisible by $k!$ (k factorial). For instance, the product of any three consecutive integers ($n(n+1)(n+2)$) is always divisible by $3! = 6$.

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2. Universal Divisibility Rules & Multi-Digit Shortcuts

Divisibility tests allow you to determine whether a large integer is divisible by a divisor without conducting time-consuming long division. The table below lists the essential rules tested on the GAT:

Advanced Composite Divisibility

To test divisibility by any composite number $C$, break $C$ into two coprime factors $a$ and $b$ (meaning $\text{GCD}(a,b) = 1$) such that $C = a \times b$. The original number is divisible by $C$ if and only if it is divisible by both $a$ and $b$.

• To test for 36: Test for $4$ and $9$ (since $\text{GCD}(4,9) = 1$).
• To test for 72: Test for $8$ and $9$ (since $\text{GCD}(8,9) = 1$).

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3. Prime Factorization, Divisor Counting & GCD / LCM Principles

According to the Fundamental Theorem of Arithmetic, every integer greater than $1$ can be uniquely represented as a product of prime powers:

$$n = p_1^{a_1} \cdot p_2^{a_2} \cdot p_3^{a_3} \dots p_k^{a_k}$$

where $p_1, p_2, \dots, p_k$ are distinct prime numbers and $a_1, a_2, \dots, a_k$ are positive integer exponents.

Formula for Total Number of Positive Factors

If the canonical prime factorization of $n$ is $p_1^{a_1} \cdot p_2^{a_2} \dots p_k^{a_k}$, the total number of positive divisors $N(n)$ is calculated by adding $1$ to each exponent and taking their product:

$$N(n) = (a_1 + 1)(a_2 + 1)(a_3 + 1) \dots (a_k + 1)$$

Worked Example of Divisor Count:

Find the total number of positive factors of $360$.

1. Execute prime factorization: $360 = 36 \times 10 = (2^2 \times 3^2) \times (2 \times 5) = 2^3 \times 3^2 \times 5^1$.
2. Apply the divisor formula: $N(360) = (3 + 1)(2 + 1)(1 + 1) = 4 \times 3 \times 2 = 24$ total factors.

Greatest Common Divisor (GCD) & Least Common Multiple (LCM)

• Greatest Common Divisor (GCD / HCF): The largest positive integer that divides two or more integers without leaving a remainder. Found by taking the lowest power of common prime factors.
• Least Common Multiple (LCM): The smallest positive integer divisible by two or more integers. Found by taking the highest power of all prime factors present.

Core GCD-LCM Theorem:

For any two positive integers $a$ and $b$:

$$a \times b = \text{GCD}(a, b) \times \text{LCM}(a, b)$$

Fractional GCD and LCM Rules:

When working with simplified fractions $\frac{a}{b}$ and $\frac{c}{d}$:

$$\text{GCD}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{GCD}(a, c)}{\text{LCM}(b, d)}$$ $$\text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)}$$

4. Cyclicity of Unit Digits & Remainder Principles

When large powers appear on the GAT (e.g., "What is the units digit of $7^{143}$?"), calculating the actual value is impossible. Instead, exploit the cyclicity of units digits under exponentiation.

Cyclicity Patterns (Base 0 through 9)

Every digit has a predictable repeating cycle of unit digits when raised to successive positive integer powers ($n = 1, 2, 3, 4, \dots$):

Step-by-Step Cyclicity Algorithm:

To find the units digit of $B^E$:

1. Identify the cycle length $C$ for base digit $B$ (use $C = 4$ as a universal cycle length for all digits).
2. Calculate the remainder $R = E \pmod C$.
3. If $R > 0$, the units digit of $B^E$ is equal to the units digit of $B^R$.
4. If $R = 0$, the units digit of $B^E$ is equal to the units digit of $B^C$.

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5. Comprehensive GAT Worked Examples

Worked Example 1: Divisibility with Unknown Digits

Question: Find the single digit $X$ such that the 6-digit number $53X,744$ is divisible by $11$.

Solution:

1. Apply the divisibility test for $11$: difference between odd-placed digits sum and even-placed digits sum.
2. Sum of odd-placed digits (from left): $5 + X + 4 = 9 + X$.
3. Sum of even-placed digits (from left): $3 + 7 + 4 = 14$.
4. Difference $= (9 + X) - 14 = X - 5$.
5. For the number to be divisible by $11$, $(X - 5)$ must equal $0$ (or a multiple of $11$).
6. Setting $X - 5 = 0 \implies X = 5$.
7. Final Answer: $X = 5$.

Worked Example 2: Units Digit of Exponential Sums

Question: What is the units digit of $7^{85} + 3^{42}$?

Solution:

1. Evaluate $7^{85}$ units digit:
   • Base $7$ cycle length is $4$: $7, 9, 3, 1$.
   • Exponent remainder: $85 \div 4 = 21$ remainder $1$.
   • So, $7^{85}$ has the same units digit as $7^1 = \mathbf{7}$.
2. Evaluate $3^{42}$ units digit:
   • Base $3$ cycle length is $4$: $3, 9, 7, 1$.
   • Exponent remainder: $42 \div 4 = 10$ remainder $2$.
   • So, $3^{42}$ has the same units digit as $3^2 = \mathbf{9}$.
3. Combine units digits: $7 + 9 = 16 \rightarrow$ units digit is $\mathbf{6}$.
4. Final Answer: $6$.

Worked Example 3: LCM Real-World Application

Question: Three traffic signals flash at intervals of $12$ seconds, $18$ seconds, and $30$ seconds respectively. If they flash together at 9:00 AM, at what time will they next flash simultaneously?

Solution:

1. Simultaneous flashing occurs at time intervals equal to the LCM of $12, 18,$ and $30$.
2. Prime factorizations:
   • $12 = 2^2 \times 3^1$
   • $18 = 2^1 \times 3^2$
   • $30 = 2^1 \times 3^1 \times 5^1$
3. $\text{LCM}(12, 18, 30) = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180$ seconds.
4. Convert seconds to minutes: $180 \text{ seconds} = 3 \text{ minutes}$.
5. Add $3$ minutes to 9:00 AM $\rightarrow$ 9:03 AM.
6. Final Answer: 9:03 AM.