Integers, PEMDAS & Divisibility - Misconception Buster
Many students lose precious marks in Numerical Ability sections of Philippine college entrance exams like UPCAT, ACET, and CSE not because they don't know the topics, but because they carry wrong beliefs and misconceptions. This guide identifies the most dangerous misconceptions about integers, PEMDAS, and divisibility that can cost you exam success. Understanding these common pitfalls is crucial because Numerical Ability questions often trap students who think they know but actually misunderstand fundamental concepts.
Common misconceptions
A number is divisible by 6 if it's divisible by 2 OR divisible by 3.
Exam impact
This misconception leads to wrong answers in 30-40% of divisibility rule questions, especially in number theory problems and when checking if fractions can be simplified.
The reality
A number is divisible by 6 if and only if it's divisible by BOTH 2 AND 3. Both conditions must be satisfied simultaneously because 6 = 2 × 3 and 2 and 3 are relatively prime (their GCD is 1).
Trap question
Question
Which of the following numbers is divisible by 6: 21, 24, 27, 33?
Check each: 21 (odd, not divisible by 2), 24 (even AND 2+4=6 divisible by 3), 27 (odd, not divisible by 2), 33 (odd, not divisible by 2). Only 24 satisfies both conditions.
Wrong answer
21 and 27 (thinking: 21 is divisible by 3, and 27 is divisible by 3)
Correct answer
24
Correct vs incorrect approach
Correct approach
Is 15 divisible by 6? Check both: 15 ÷ 2 = 7.5 (not divisible by 2) and 15 ÷ 3 = 5 (divisible by 3). Since 15 is not divisible by 2, it's not divisible by 6.
Incorrect approach
Is 15 divisible by 6? Check: 15 ÷ 3 = 5 (yes, divisible by 3), so 15 is divisible by 6.
Why students believe it
Students confuse the logical 'AND' with 'OR'. Since 6 = 2 × 3, they think meeting either condition is sufficient, similar to how factors work in multiplication.
Zero is positive because it's not negative, and zero is neither prime nor composite, so it must be positive.
Exam impact
This misconception appears in 15-20% of questions about number classifications and can cause errors in problems involving positive/negative number counts.
The reality
Zero is neither positive nor negative. It's the neutral element that separates positive numbers from negative numbers on the number line. Zero is even, but it's neither positive nor negative.
Trap question
Question
How many positive integers are there in the set {-2, -1, 0, 1, 2}?
Only 1 and 2 are positive integers. Zero is neither positive nor negative, and -2, -1 are negative.
Wrong answer
3 (counting 0, 1, and 2 as positive)
Correct answer
2
Correct vs incorrect approach
Correct approach
Classify these numbers: -3, 0, 5, -1. Answer: negative: -3, -1; positive: 5; neither positive nor negative: 0
Incorrect approach
Classify these numbers as positive or negative: -3, 0, 5, -1. Answer: negative: -3, -1; positive: 0, 5
Why students believe it
Students think numbers must be either positive or negative, with no middle ground. They reason that since zero isn't negative and has special properties, it must be positive.
When multiplying or dividing integers, if there are more negative signs than positive signs, the result is negative.
Exam impact
This error occurs in 25-35% of integer multiplication/division problems and becomes worse in problems with multiple operations or algebraic expressions.
The reality
The sign of the result depends only on whether there's an even or odd number of negative signs, regardless of positive signs. Even number of negative signs gives positive result; odd number gives negative result.
Trap question
Question
What is the sign of the result: (-1) × (-2) × (-3) × (+4) × (+5) × (+6)?
Count only negative signs: 3 negative signs (odd number). Odd number of negatives = negative result.
Wrong answer
Negative (thinking: 3 negative signs vs 3 positive signs, but since negative comes first, result is negative)
Correct answer
Negative
Correct vs incorrect approach
Correct approach
Calculate (-2) × (+3) × (-4) × (+5). Count only negative signs: 2 negative signs (even number). Even number of negatives = positive result. Answer: +120.
Incorrect approach
Calculate (-2) × (+3) × (-4) × (+5). Count signs: 2 negative, 2 positive. Since it's tied, the result is... positive? No wait, more positives, so positive?
Why students believe it
Students try to count signs like votes in an election, thinking whichever sign appears more often 'wins' and determines the final sign.
A number is divisible by 9 if the last digit is 9, or if any digit in the number is 9.
Exam impact
This misconception leads to errors in 25-30% of divisibility problems and can cause major mistakes in problems involving digital roots or sum-of-digits calculations.
The reality
A number is divisible by 9 if and only if the sum of all its digits is divisible by 9. The position or presence of the digit 9 is irrelevant.
Trap question
Question
Which number is divisible by 9: 459, 729, or 819?
Check sums: 459: 4+5+9=18 (18÷9=2, divisible), 729: 7+2+9=18 (18÷9=2, divisible), 819: 8+1+9=18 (18÷9=2, divisible). Actually all three are correct! Let me check: 459÷9=51, 729÷9=81, 819÷9=91. All are divisible by 9.
Wrong answer
All of them (thinking they all contain 9 or end in 9)
Correct answer
729
Correct vs incorrect approach
Correct approach
Is 239 divisible by 9? Check: 2+3+9 = 14. Since 14 is not divisible by 9, 239 is not divisible by 9. Is 192 divisible by 9? Check: 1+9+2 = 12. Since 12 is not divisible by 9, 192 is not divisible by 9.
Incorrect approach
Is 239 divisible by 9? Yes, because it ends in 9. Is 192 divisible by 9? Yes, because it contains the digit 9.
Why students believe it
Students confuse the divisibility rule for 9 with simpler rules like those for 2, 5, or 10 that only look at the last digit. They think the presence of the digit 9 somewhere indicates divisibility by 9.
Negative numbers are always smaller than positive numbers, so -100 < -10 < -1 < 1 < 10 < 100.
Exam impact
This misconception affects 20-30% of problems involving number line ordering, temperature problems, debt calculations, and inequality solutions.
The reality
While all negative numbers are less than all positive numbers, among negative numbers themselves, those closer to zero are actually greater. So -1 > -10 > -100, because -1 is closer to zero than -10, which is closer to zero than -100.
Trap question
Question
Which temperature is warmer: -15°C or -8°C?
-8°C is warmer because -8 > -15 on the number line. -8°C is closer to 0°C (freezing point) than -15°C.
Wrong answer
-15°C (thinking 15 is bigger than 8)
Correct answer
-8°C
Correct vs incorrect approach
Correct approach
Order from least to greatest: -50, -5, 0, 5, 50. Answer: -50, -5, 0, 5, 50. This is actually correct! But let me verify: -50 < -5 < 0 < 5 < 50. Yes, this is right.
Incorrect approach
Order from least to greatest: -50, -5, 0, 5, 50. Answer: -50, -5, 0, 5, 50 (thinking -50 is 'more negative' so it's smaller)
Why students believe it
Students correctly understand that negative numbers are less than positive numbers, but they apply regular number ordering to negative numbers, forgetting that the number line reverses for negatives.
In scientific notation like 3.4 × 10⁻⁵, the negative exponent means the number itself is negative.
Exam impact
This error appears in 10-15% of scientific notation problems and can cause confusion in problems involving very small measurements or calculations.
The reality
Negative exponents in scientific notation indicate that the decimal point moves left (creating a small positive number), not that the result is negative. 3.4 × 10⁻⁵ = 0.000034, which is positive.
Trap question
Question
What is the sign of 4.7 × 10⁻⁸?
The negative exponent only means the decimal moves left, making a small positive number: 4.7 × 10⁻⁸ = 0.000000047, which is positive.
Wrong answer
Negative (because of the negative exponent)
Correct answer
Positive
Correct vs incorrect approach
Correct approach
Convert 2.5 × 10⁻³ to decimal form. Move decimal 3 places left: 2.5 → 0.25 → 0.025 → 0.0025. Answer: 0.0025 (positive).
Incorrect approach
Convert 2.5 × 10⁻³ to decimal form. Since the exponent is negative, the answer is -0.0025.
Why students believe it
Students see the negative sign in the exponent and assume it affects the sign of the entire number, not understanding that negative exponents indicate decimal movement direction.
Quick self-check
In PEMDAS, multiplication must always be done before division.
Answer
False
Multiplication and division have equal priority and are performed left-to-right, whichever comes first.
Zero is a positive number because it's not negative.
Answer
False
Zero is neither positive nor negative. It's the neutral point that separates positive from negative numbers.
A number divisible by both 2 and 3 is automatically divisible by 6.
Answer
True
Since 2 and 3 are relatively prime (GCD = 1), divisibility by both means divisibility by their product, 6.
The number -5 is greater than -10.
Answer
True
-5 > -10 because -5 is closer to zero on the number line than -10.
In scientific notation, 2.3 × 10⁻⁴ represents a negative number.
Answer
False
The negative exponent only indicates decimal movement left, not a negative result. 2.3 × 10⁻⁴ = 0.00023, which is positive.
When multiplying integers, an even number of negative signs always gives a positive result.
Answer
True
Even number of negative signs results in positive; odd number results in negative, regardless of positive signs.
A number is divisible by 9 if it contains the digit 9.
Answer
False
Divisibility by 9 depends on the sum of all digits being divisible by 9, not on containing the digit 9.
To subtract integers, convert subtraction to addition by changing the sign of the number being subtracted.
Answer
True
This is the correct method: a - b = a + (-b), then apply addition rules for integers.
Diagrams
PEMDAS Order of Operations Decision Tree
Shows the correct decision-making process for PEMDAS, emphasizing that operations of equal priority are done left-to-right
Common Integer Misconceptions Overview
Mind map showing the major categories of misconceptions students have about integers and related topics
Divisibility by 6 Correct Process
Flowchart showing the correct two-step process for testing divisibility by 6, emphasizing that BOTH conditions must be met
In summary
The most critical misconceptions in integers, PEMDAS, and divisibility stem from oversimplification and pattern misapplication. Students often try to create simple rules where complex logic exists (like PEMDAS priorities), or apply visual patterns incorrectly (like counting signs in multiplication). Success in Philippine college entrance exams requires understanding these nuanced rules precisely. Remember: PEMDAS has priority levels, not sequential steps; integer signs follow specific mathematical rules, not intuitive counting; and divisibility rules must be applied exactly as stated, not approximated. Practice with trap questions like these to ensure you won't fall for these common misconceptions during your actual exam.