5.1 Number Sense and Operations
Key Takeaways
- Mathematics is about 21% of the PUPCET, and fast, accurate arithmetic is the foundation of almost every math item
- GEMDAS ranks Grouping symbols first, then Exponents, then Multiplication/Division left to right, then Addition/Subtraction left to right
- 1 is neither prime nor composite, and 2 is the only even prime number
- GCF uses the lowest powers of shared prime factors; LCM uses the highest powers of all prime factors present
- For any two numbers, GCF × LCM equals the product of the two numbers — a fast way to find one when you know the other
Why Number Sense Matters on the PUPCET
Mathematics makes up about 21% of the PUPCET — roughly 31 out of 150 multiple-choice items — and almost every math question assumes you can compute quickly and correctly without a calculator (calculators are not allowed). Number sense questions also appear disguised inside word problems in Science and General Information, so shaky fundamentals cost you points across the whole exam, not just in the math subtest.
The Family of Real Numbers
The real numbers include every number you will meet on the PUPCET. Know how the sets nest inside one another:
| Set | Description | Examples |
|---|---|---|
| Natural (counting) numbers | Numbers used for counting | 1, 2, 3, 4, ... |
| Whole numbers | Natural numbers plus zero | 0, 1, 2, 3, ... |
| Integers | Whole numbers and their negatives | ..., -2, -1, 0, 1, 2, ... |
| Rational numbers | Any number expressible as a fraction p/q where q ≠ 0; decimals that terminate or repeat | 3/4 = 0.75, 1/3 = 0.333..., -7, 0.125 |
| Irrational numbers | Cannot be written as a fraction; decimals never terminate and never repeat | π, √2, √3 |
| Real numbers | All rational and irrational numbers combined | All of the above |
A classic PUPCET trap: every integer is rational (because 5 = 5/1), but not every rational number is an integer. Zero is an integer and is rational, but it is neither positive nor negative.
Properties of Real Numbers
These properties let you rearrange computations legally — and tell you which rearrangements are illegal.
| Property | Rule | Example |
|---|---|---|
| Commutative | a + b = b + a; a × b = b × a | 3 + 5 = 5 + 3 |
| Associative | (a + b) + c = a + (b + c) | (2 + 3) + 4 = 2 + (3 + 4) |
| Distributive | a(b + c) = ab + ac | 4(5 + 2) = 20 + 8 = 28 |
| Identity | a + 0 = a; a × 1 = a | 17 × 1 = 17 |
| Inverse | a + (-a) = 0; a × (1/a) = 1 | 6 × (1/6) = 1 |
Warning: subtraction and division are not commutative. 8 − 5 ≠ 5 − 8, and 12 ÷ 4 ≠ 4 ÷ 12. The distributive property is the workhorse for mental math: 7 × 98 = 7(100 − 2) = 700 − 14 = 686.
Order of Operations: GEMDAS
When an expression mixes operations, evaluate in this order:
- G — Grouping symbols: parentheses ( ), brackets [ ], braces { }, and fraction bars (innermost first)
- E — Exponents and roots
- MD — Multiplication and Division, whichever comes first from left to right
- AS — Addition and Subtraction, whichever comes first from left to right
You may also see this called PEMDAS (Parentheses, Exponents...). Same rule, same result.
Worked Example 1: Evaluate 3 + 4 × 2² − (8 ÷ 2)
- Grouping first: (8 ÷ 2) = 4 → 3 + 4 × 2² − 4
- Exponent: 2² = 4 → 3 + 4 × 4 − 4
- Multiply: 4 × 4 = 16 → 3 + 16 − 4
- Add and subtract left to right: 19 − 4 = 15
Worked Example 2 (the classic trap): Evaluate 24 ÷ 4 × 2
Multiplication and division have equal rank, so go left to right: 24 ÷ 4 = 6, then 6 × 2 = 12. Students who multiply first get 24 ÷ 8 = 3 — a wrong answer that the test writers absolutely include as a distractor.
Worked Example 3: Evaluate (7 + 5)² ÷ (9 − 3)
- Groupings: (12)² ÷ (6)
- Exponent: 144 ÷ 6 = 24
Factors, Multiples, and Primes
A factor of a number divides it exactly (the factors of 12 are 1, 2, 3, 4, 6, 12). A multiple of a number is the product of that number and an integer (multiples of 12 are 12, 24, 36, ...). A prime number has exactly two factors — 1 and itself — while a composite number has more than two. Memorize this: 1 is neither prime nor composite, and 2 is the only even prime.
Primes under 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Know this list cold.
Divisibility Rules
These let you factor numbers in seconds without long division:
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 4,536 ends in 6 ✓ |
| 3 | Digit sum is divisible by 3 | 4,536: 4+5+3+6 = 18 ✓ |
| 4 | Last two digits divisible by 4 | 4,536: 36 ÷ 4 = 9 ✓ |
| 5 | Ends in 0 or 5 | 1,245 ✓ |
| 6 | Divisible by both 2 and 3 | 4,536 ✓ |
| 8 | Last three digits divisible by 8 | 7,128: 128 ÷ 8 = 16 ✓ |
| 9 | Digit sum is divisible by 9 | 4,536: 18 ÷ 9 = 2 ✓ |
| 10 | Ends in 0 | 3,270 ✓ |
| 11 | Alternating digit sum divisible by 11 | 9,284: (9−2+8−4) = 11 ✓ |
GCF and LCM
The Greatest Common Factor (GCF) is the largest number that divides two or more numbers exactly. The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers.
Method 1 — Listing. GCF of 36 and 48: factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36; factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The largest shared factor is 12.
Method 2 — Prime factorization. 36 = 2² × 3² and 48 = 2⁴ × 3. For the GCF, take each shared prime at its lowest power: 2² × 3 = 12. For the LCM, take every prime that appears at its highest power: 2⁴ × 3² = 16 × 9 = 144.
Handy identity: GCF(a, b) × LCM(a, b) = a × b. Check: 12 × 144 = 1,728 = 36 × 48. If a problem gives you the GCF and asks for the LCM, this identity is your shortcut.
Word-Problem Applications
LCM = repeating events line up. Three shuttle buses leave the PUP Sta. Mesa terminal every 12, 18, and 20 minutes. If they all depart together at 6:00 AM, when do they next leave together? Compute LCM(12, 18, 20): 12 = 2² × 3, 18 = 2 × 3², 20 = 2² × 5, so LCM = 2² × 3² × 5 = 180 minutes = 3 hours, or 9:00 AM.
GCF = largest equal grouping. A teacher wants to group 36 boys and 48 girls into the largest possible number of identical teams, each with the same mix of boys and girls. GCF(36, 48) = 12 teams, each with 3 boys and 4 girls.
Traps and Exam Tips
- Questions saying smallest, soonest, or when will they meet again usually need the LCM; largest equal groups or greatest possible size usually need the GCF.
- 0 is a multiple of every number, but a factor of no number.
- Do not confuse factor (divides the number, smaller or equal) with multiple (the number divides it, larger or equal).
- On GEMDAS items, write one step per line on your scratch paper — most errors come from doing two steps mentally at once.
Evaluate: 24 − 6 ÷ 2 × 3 + 4
Three jeepneys leave a terminal together at 8:00 AM. One returns every 12 minutes, another every 18 minutes, and the third every 20 minutes. In how many minutes will all three be at the terminal together again?