5.1 Divisibility, Primes & Number Properties

Arithmetic & Number Properties on the GAT

Arithmetic forms the foundation of the Quantitative Reasoning section in the Saudi Arabia Qiyas General Aptitude Test (GAT / Qudurat). GAT arithmetic questions prioritize patterns, algebraic logic, and shortcuts over tedious calculations. This section covers integers, divisibility rules, prime factorization, greatest common divisors (GCD), least common multiples (LCM), even/odd algebra, and modular digit cycles.

Integers & Sign Operations

In the GAT, numbers are categorized as integers, fractions, decimals, or reals. Integers (Z) consist of positive whole numbers, negative whole numbers, and zero (0). Zero (0) is a frequently tested integer on the GAT. It is even, but it is neither positive nor negative. Multiplying any number by zero results in zero. Division by zero is undefined. Zero is also a multiple of every integer. For signs: Positive * Positive = Positive; Negative * Negative = Positive; Positive * Negative = Negative. Even and odd powers of negative numbers: (-x)^n yields a positive value if n is even, and a negative value if n is odd. Always distinguish between (-3)^2 = 9 and -3^2 = -9, as GAT questions often exploit this difference.

Divisibility Rules & Shortcuts

To solve questions quickly, you must recognize divisibility without executing long division:

• Divisibility by 2: The units digit is even (0, 2, 4, 6, 8).
• Divisibility by 3: Digits sum to a multiple of 3. Example: 1,437 (1+4+3+7=15).
• Divisibility by 4: Last two digits form a multiple of 4. Example: 5,824.
• Divisibility by 5: The units digit is either 0 or 5.
• Divisibility by 6: Divisible by 2 and 3. Example: 132.
• Divisibility by 8: Last three digits form a multiple of 8. Example: 4,128.
• Divisibility by 9: Digits sum to a multiple of 9. Example: 7,308.
• Divisibility by 10: The units digit is 0.
• Divisibility by 11: Alternating sum of digits is a multiple of 11. Example: 8,294 ((8+9)-(2+4)=11).

Prime Numbers & Factor Counting

A prime number is an integer greater than 1 with exactly two divisors: 1 and itself. The number 2 is the only even prime; 1 is not prime. To calculate the total number of positive factors of an integer, write its prime factorization: N = p^a * q^b * r^c. The number of positive factors is: (a + 1)(b + 1)(c + 1). Example: How many positive divisors does 72 have? Prime factorization is 72 = 2^3 * 3^2. Total factors = (3 + 1)(2 + 1) = 12.

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

Greatest Common Divisor (GCD) is the largest integer dividing the numbers without remainder. Multiply the lowest powers of common prime factors. Example: For 48 (2^4 * 3^1) and 180 (2^2 * 3^2 * 5^1), GCD is 2^2 * 3^1 = 12. Least Common Multiple (LCM) is the smallest positive common multiple. Multiply the highest powers of all prime factors. Example: LCM of 48 and 180 is 2^4 * 3^2 * 5^1 = 720.

Word Scenarios:

• LCM: If Bus A departs every 12 minutes and Bus B every 18 minutes, starting together at 8:00 AM, they next depart together at the LCM of 12 and 18, which is 36 minutes later (8:36 AM).
• GCD: To divide 24 pens and 36 notebooks into identical bags with no leftovers, the maximum number of bags is the GCD of 24 and 36, which is 12.

Even and Odd Integers Algebra

GAT operations on odd (O) and even (E) integers follow strict rules:

• Addition/Subtraction: E + E = E; E + O = O; O + O = E.
• Multiplication: E * E = E; E * O = E; O * O = O.
• Exponentiation: E^n is even; O^n is odd. Strategy: If a question asks which expression must be odd given odd x and even y, substitute x = 1 and y = 2 to evaluate options immediately, avoiding algebraic errors.

Power Cycle Patterns for Units Digits

GAT frequently tests your ability to find the units digit of high powers. Focus only on the units digit of the base and use its cyclicity length: 0, 1, 5, and 6 have a cycle of 1. 4 and 9 have a cycle of 2. 2, 3, 7, and 8 have a cycle of 4. Divide the exponent by the cycle length to find the remainder. A remainder of 1, 2, or 3 corresponds to that position in the cycle; a remainder of 0 corresponds to the last position. Worked Example: What is the units digit of 7^98? The base is 7 (cycle: 7, 9, 3, 1). Divide 98 by 4: 98 / 4 = 24 remainder 2. Since the remainder is 2, select the second term in the cycle, which is 9.

Consecutive Integers & Sum Sequences

Consecutive integers are whole numbers that follow in sequence, each 1 unit larger than the last.

• Average-Median Rule: For any arithmetic sequence, the average is exactly equal to the median (middle term). Average = (First Term + Last Term) / 2.
• Sum of Sequence: Sum = Average * Number of Terms.
• Sum of first n positive integers: Sum = n(n+1) / 2.

Common Qiyas Arithmetic Traps

• The Prime Number 1 Trap: GAT questions frequently include 1 in lists of numbers to test if you recognize it as non-prime. Always exclude 1 from prime lists.
• The Even Zero Trap: Remember, 0 is an even integer. If the problem specifies 'positive integers,' 0 is excluded. If it just says 'integers,' 0 is included.
• Digit Sum vs Product: In divisibility, do not confuse the rule of 3 (sum of digits) with the rule of 4 (last two digits). Make sure to practice the alternate sum rule for 11 carefully.