4.1 Basic Operations & Number Properties

Basic Operations & Number Properties

The Numerical Ability section of the Civil Service Examination – Pen and Paper Test (CSE-PPT) Subprofessional Level tests whether you can compute quickly and accurately without a calculator. Roughly a quarter of your total score depends on this chapter, and every other numerical topic in this study guide — fractions, word problems, and series — assumes you have mastered the basics covered here: the correct order of operations, working confidently with positive and negative whole numbers, and recognizing key number properties like factors, multiples, primes, and divisibility.

Order of Operations (MDAS / GMDAS)

When an expression has more than one operation, you cannot simply compute left to right. You must follow a fixed sequence, often taught in Philippine schools as GMDAS (Grouping symbols, Multiplication and Division, Addition and Subtraction) or the more familiar PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Both describe the same rule:

1. Grouping symbols / Parentheses — solve everything inside ( ), [ ], or { } first.
2. Exponents — apply any powers or roots.
3. Multiplication and Division — performed together, strictly left to right, in the order they appear.
4. Addition and Subtraction — performed together, strictly left to right, in the order they appear.

A common mistake is assuming multiplication always comes before division, or addition always before subtraction. It does not — within each pair, you work left to right.

Worked Example 1

Evaluate: 12 + 3 × (8 − 5)² ÷ 9

• Step 1 (grouping): 8 − 5 = 3, so the expression becomes 12 + 3 × 3² ÷ 9.
• Step 2 (exponent): 3² = 9, giving 12 + 3 × 9 ÷ 9.
• Step 3 (multiplication/division, left to right): 3 × 9 = 27, then 27 ÷ 9 = 3.
• Step 4 (addition): 12 + 3 = 15.

Answer: 15

Worked Example 2

Evaluate: (−7) × (−4) − 20 ÷ 4 + 3²

• Step 1 (exponent): 3² = 9.
• Step 2 (multiplication/division, left to right): (−7) × (−4) = 28; 20 ÷ 4 = 5.
• Step 3 (addition/subtraction, left to right): 28 − 5 + 9 = 32.

Answer: 32

Working with Integers and Negative Numbers

Integers are whole numbers that can be positive, negative, or zero (…, −3, −2, −1, 0, 1, 2, 3, …). CSE-PPT questions frequently mix signed numbers, so memorize these rules:

• Addition: Same signs — add the magnitudes and keep the sign (−5 + (−3) = −8). Different signs — subtract the smaller magnitude from the larger and keep the sign of the larger magnitude (−9 + 4 = −5, because 9 is larger and negative).
• Subtraction: Change the subtraction to addition of the opposite, then apply the addition rule (6 − (−2) = 6 + 2 = 8).
• Multiplication and Division: Same signs produce a positive result; different signs produce a negative result ((−6) × (−3) = 18; (−6) × 3 = −18; 20 ÷ (−4) = −5).

Worked Example 3

Evaluate: (−9) × 4 ÷ (−6) + (−3)

• Step 1 (multiplication/division, left to right): (−9) × 4 = −36.
• Step 2: −36 ÷ (−6) = 6 (negative divided by negative is positive).
• Step 3 (addition): 6 + (−3) = 3.

Answer: 3

Factors, Multiples, and Prime Numbers

A factor of a number divides into it evenly, with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. A multiple of a number is the product of that number and any whole number — the multiples of 4 are 4, 8, 12, 16, 20, and so on, continuing forever.

A prime number has exactly two factors: 1 and itself (examples: 2, 3, 5, 7, 11, 13, 17, 19, 23...). Note that 2 is the only even prime number — every other even number is divisible by 2, so it automatically has at least three factors. A composite number has more than two factors (4, 6, 8, 9, 10, 12...). The number 1 is neither prime nor composite.

The greatest common factor (GCF) is the largest number that divides two or more numbers evenly (the GCF of 12 and 18 is 6). The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers (the LCM of 4 and 6 is 12). Both concepts reappear when you add or subtract fractions in the next section.

Divisibility Rules

Divisibility rules let you check whether a number divides evenly into another without doing long division — a major time-saver on a timed exam.

Worked Example 4

Is 4,536 divisible by 2, 3, 4, 5, 6, 9, and 10?

Using the table above: 4,536 is divisible by 2 (ends in 6), 3 and 9 (digit sum 18), 4 (last two digits 36 ÷ 4 = 9), and 6 (divisible by both 2 and 3). It is not divisible by 5 or 10, because it does not end in 0 or 5.

Exponents and Square Roots

An exponent tells you how many times to multiply a number by itself: 5² = 5 × 5 = 25 and 2³ = 2 × 2 × 2 = 8. A square root (√) asks the reverse question — what number, multiplied by itself, gives this result? √49 = 7 because 7 × 7 = 49.

Memorize the perfect squares from 1² through 15² (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225) — recognizing them instantly saves time on the exam. When a square root is not a perfect square, estimate using the two nearest perfect squares: √50 falls between √49 = 7 and √64 = 8, and since 50 is closer to 49, √50 ≈ 7.1.

Exam Strategy

On test day, rewrite each expression step by step rather than solving in your head — miswritten signs are the most common source of errors. Circle grouping symbols first, and double-check any subtraction of negative numbers, since that single step causes the majority of mistakes on this topic.