Integers, PEMDAS & Divisibility - Memory Anchors
Memory techniques are powerful tools that transform abstract mathematical concepts into unforgettable mental images and stories. By using mnemonics, analogies, and visual associations, you can dramatically improve recall speed and accuracy - essential skills for college entrance exams like UPCAT and CSE. These memory anchors create multiple pathways to the same information, making it nearly impossible to forget key concepts during high-pressure exam situations.
Memory anchors
PEMDAS Order of Operations
Memory aid
Please Excuse My Dear Aunt Sally - or in Filipino context: 'Paki Excuse Mo Dear Ate Sally'
Why it works
The acronym creates a memorable sentence that maps directly to the order: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
Example usage
For 2 + 3 × 4², think 'Ate Sally': First exponents (4² = 16), then multiplication (3 × 16 = 48), finally addition (2 + 48 = 50)
Recall trigger
When you see a complex expression, think 'Ate Sally' to remember the order
Negative number addition rules
Memory aid
Adding negative numbers is like debt and money. Same sign friends stick together and add up their values. Opposite sign friends fight, and the stronger one wins but loses some strength.
Why it works
The debt/money analogy makes abstract positive/negative relationships concrete and relatable
Example usage
For (-5) + (-3), both are debts so they stick together: -8. For (-5) + 8, debt fights money, money wins: +3
Recall trigger
Think 'debt vs money' when adding integers with different signs
Multiplication of integers sign rules
Memory aid
Same signs make it positive, Different signs make it negative. Like attracts like, opposites fight!
Why it works
The rhyme creates a musical pattern that's easy to remember, and the 'like attracts like' concept is intuitive
Example usage
For (-3) × (-4), same signs (both negative) = positive 12. For (-3) × 4, different signs = negative 12
Recall trigger
When multiplying integers, think 'like attracts like'
Divisibility by 2
Memory aid
Picture a pair of shoes - everything even can be paired up perfectly. If the last digit is even (0,2,4,6,8), it's divisible by 2, like shoes that come in pairs.
Why it works
Visual imagery of pairs connects to the concept of even numbers being divisible by 2
Example usage
For 1,234, look at the last digit (4). Think shoes - yes, 4 shoes make 2 pairs, so 1,234 is divisible by 2
Recall trigger
See the last digit, think 'can it pair up like shoes?'
Divisibility by 3
Memory aid
Three friends (3, 6, 9) love to share everything equally. To check if a number likes them, add up all its digits. If that sum is a friend of 3 (divisible by 3), then the original number will share with them too.
Why it works
The story personifies numbers and creates a logical connection between digit sum and divisibility
Example usage
For 576: 5+7+6=18. Since 18 likes the three friends (18÷3=6), so does 576
Recall trigger
Think 'three friends sharing' when checking divisibility by 3
Divisibility by 5
Memory aid
Five pesos coin and ten pesos coin - money that ends in 5 or 0. Any number ending in 5 or 0 is 'valuable' to 5 (divisible).
Why it works
Connects to familiar Philippine currency and the visual pattern of 5 and 0 endings
Example usage
For 1,275, it ends in 5 like a 5-peso coin, so it's divisible by 5
Recall trigger
Think 'pesos ending in 5 or 0' when checking divisibility by 5
Divisibility by 9
Memory aid
NINE DINES when the digit sum is fine (divisible by 9). Nine is very picky - only numbers whose digit sum is also divisible by 9 can join its dinner table.
Why it works
The rhyme 'nine dines when digit sum is fine' creates a memorable rule pattern
Example usage
For 729: 7+2+9=18, and 18÷9=2, so 729 can join nine's dinner table (is divisible by 9)
Recall trigger
Think 'nine's picky dinner table' when checking divisibility by 9
Scientific notation structure
Memory aid
Scientific notation is like a moving truck: the mantissa (1-10) is the cargo that stays the same size, and the exponent is the number of houses you move left or right on the number street.
Why it works
The moving analogy helps visualize decimal point movement and maintains the relationship between parts
Example usage
For 45,000: move the decimal 4 houses right to get 4.5, so it's 4.5 × 10⁴
Recall trigger
Think 'moving truck on number street' when converting to scientific notation
Converting positive to negative exponents
Memory aid
Positive exponents are like going UP a building (bigger numbers), negative exponents are like going DOWN to the basement (smaller numbers, towards zero). Moving the decimal left goes up, moving right goes down.
Why it works
The up/down building metaphor makes the inverse relationship between decimal movement and exponent signs clear
Example usage
For 0.0045: move decimal right 3 places to get 4.5, so we went 'down' 3 floors: 4.5 × 10⁻³
Recall trigger
Think 'building floors' when dealing with positive/negative exponents
Absolute value definition
Memory aid
Absolute value is like a mirror - it only shows positive reflections. No matter if you're happy (+) or sad (-), your reflection in the absolute value mirror is always positive distance from zero.
Why it works
The mirror metaphor makes the 'always positive' concept intuitive and memorable
Example usage
For |-7|, imagine -7 looking in the absolute value mirror and seeing +7 looking back
Recall trigger
Think 'positive mirror reflection' when seeing absolute value bars |n|
Zero properties in operations
Memory aid
Zero's Seven Special Powers: Addition (no change), Subtraction (no change), Multiplication (becomes zero), Division by zero (forbidden/undefined), Divided by others (becomes zero), Exponent (makes one), Any number to power zero (equals one).
Why it works
Chunking the seven properties into 'powers' makes them easier to memorize as a group
Example usage
For 5 × 0, think zero's multiplication power - everything becomes zero. For 5⁰, think zero exponent power - becomes one
Recall trigger
Think 'Zero's Seven Special Powers' when dealing with zero in operations
Prime vs Composite numbers
Memory aid
Prime numbers are like introverted people - they only have two real friends (1 and themselves). Composite numbers are like popular people at parties - they have many factor friends who can multiply together to make them.
Why it works
The personality analogy makes the mathematical distinction relatable and memorable
Example usage
For 17, it's introverted (only friends: 1 and 17), so it's prime. For 12, it's popular (friends: 1,2,3,4,6,12), so it's composite
Recall trigger
Think 'introvert vs popular person' when classifying numbers as prime or composite
Finding GCF using prime factorization
Memory aid
Imagine a factory assembly line: First station breaks numbers into prime parts, Second station compares the parts from both numbers, Third station picks only the common parts, Fourth station multiplies common parts together for the final GCF product.
Why it works
The assembly line creates a spatial memory sequence that maps to the mathematical process
Example usage
For GCF of 12 and 18: Factory breaks them (12=2²×3, 18=2×3²), compares parts, picks common ones (2¹×3¹), final product: GCF=6
Recall trigger
Think 'GCF factory assembly line' when finding greatest common factors
Divisibility by 4
Memory aid
Four wheels on a car - look at the last TWO digits like the back two wheels. If those two digits can be divided by 4 evenly, the whole car (number) can roll smoothly with 4.
Why it works
The car wheel visual connects the concept of looking at last two digits with the rule for divisibility by 4
Example usage
For 1,236, look at the back wheels (36). Since 36÷4=9 exactly, the car 1,236 rolls smoothly with 4
Recall trigger
Think 'back two wheels of a car' when checking divisibility by 4
LCM using prime factorization
Memory aid
The LCM Hotel needs to accommodate all guests (numbers) comfortably. The hotel manager takes the HIGHEST power of each prime factor from all numbers to ensure everyone fits. It's like booking the biggest room size that each family needs.
Why it works
The hotel story explains why we take the highest powers - to accommodate all requirements
Example usage
For LCM of 12 and 18: Hotel needs 2² rooms (for 12's family) and 3² rooms (for 18's family), so LCM = 2² × 3² = 36
Recall trigger
Think 'LCM Hotel with biggest rooms' when finding least common multiple
Converting fractions to decimals to percentages
Memory aid
FDP Chain: Fraction Divides to Decimal, then Decimal × 100 to Percentage. Remember: F→D (divide), D→P (×100), P→D (÷100), D→F (make fraction)
Why it works
The chain acronym creates a clear conversion pathway between all three forms
Example usage
For ¾: F→D (3÷4=0.75), D→P (0.75×100=75%). Reverse: 75%→D (75÷100=0.75), D→F (75/100=¾)
Recall trigger
Think 'FDP Chain' when converting between fractions, decimals, and percentages
Divisibility by 6
Memory aid
SIX = Sum (divisible by 3) + eXtra (even number). For a number to be divisible by 6, it must pass both the 3-test and 2-test.
Why it works
The acronym breaks down the compound rule into memorable components
Example usage
For 342: Sum test (3+4+2=9, divisible by 3 ✓), eXtra test (ends in 2, even ✓), so 342 is divisible by 6
Recall trigger
Think 'SIX = Sum + eXtra' when checking divisibility by 6
Order of operations with nested grouping symbols
Memory aid
Nested groupings are like Russian nesting dolls (matryoshka) - you must open the innermost doll first, then work your way outward. Parentheses ( ) are the tiny inner doll, brackets [ ] are the middle doll, braces { } are the outer doll.
Why it works
The nesting doll analogy perfectly visualizes the inside-out approach to nested groupings
Example usage
For {5 + [3 × (2 + 1)]}: Open inner doll (2+1=3), middle doll [3×3=9], outer doll {5+9=14}
Recall trigger
Think 'Russian nesting dolls - innermost first' when seeing complex groupings
Divisibility by 11
Memory aid
Eleven is like a seesaw - alternating positions must balance. Add digits in odd positions, add digits in even positions, then check if their difference is 0 or divisible by 11. If balanced, the number works with 11.
Why it works
The seesaw visual helps remember the alternating pattern and balance requirement
Example usage
For 4,510: Odd positions (4+1=5), Even positions (5+0=5), Difference (5-5=0). Seesaw balanced, so divisible by 11
Recall trigger
Think 'seesaw balance' when checking divisibility by 11
Properties of multiples
Memory aid
Multiples are magic, here's what they do: Every number's a multiple of itself and one too! The smallest multiple is the number you see, They go on forever, infinitely free!
Why it works
The rhyme encodes the four key properties in a memorable, musical pattern
Example usage
For multiples of 7: 7 is multiple of itself (7×1=7), smallest is 7, they continue forever: 7,14,21,28...
Recall trigger
Think 'multiples are magic' when listing properties of multiples
Revision game
I'm the operation that comes first in PEMDAS, you'll find me wrapped in curved lines
Answer
Parentheses
Memory link
Please Excuse My Dear Ate Sally mnemonic (A1)
I only have two factors - myself and one. I'm quite antisocial in the number world!
Answer
Prime number
Memory link
Introverted people analogy (A12)
To test if I work with 3, add up all my digits. If that sum likes 3, then so do I!
Answer
Divisibility by 3
Memory link
Three friends sharing story (A5)
I'm like a mirror that only shows positive reflections, no matter what looks at me
Answer
Absolute value
Memory link
Positive mirror reflection analogy (A10)
When same signs multiply, they make friends and stay positive. When different signs multiply, they fight!
Answer
Integer multiplication rules
Memory link
Like attracts like rhyme (A3)
I need both even last digit AND digit sum divisible by 3. I'm quite demanding!
Answer
Divisibility by 6
Memory link
SIX = Sum + eXtra acronym (A17)
I'm like Russian nesting dolls - you must open the innermost one first
Answer
Nested grouping symbols
Memory link
Matryoshka dolls analogy (A18)
I'm the hotel manager who needs the biggest room size to accommodate all number families
Answer
LCM (Least Common Multiple)
Memory link
LCM Hotel story (A15)
Formula mnemonics
PEMDAS: P-E-M/D-A/S
Mnemonic
Please Excuse My Dear Ate Sally (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)
When to use
When evaluating any complex mathematical expression with multiple operations
What each part means
P=Parentheses first, E=Exponents second, M/D=Multiplication and Division left to right, A/S=Addition and Subtraction left to right
Scientific Notation: a × 10^n (where 1 ≤ |a| < 10)
Mnemonic
Move decimal to make Mantissa between 1-10, Count moves for eXponent. Left moves = positive, Right moves = negative
When to use
When expressing very large or very small numbers in compact form
What each part means
a=mantissa (coefficient between 1-10), 10=base, n=exponent (number of decimal places moved)
|n| = n if n≥0, |n| = -n if n<0
Mnemonic
Absolute value Always Positive - mirror reflects only positive distances
When to use
When finding distance from zero or removing negative signs
What each part means
The absolute value bars |n| always give the positive distance from zero
GCF using prime factors: multiply all common factors with lowest powers
Mnemonic
Greatest Common Factor = Common factors with Least powers (GCF = CLp)
When to use
When finding the largest number that divides two or more numbers evenly
What each part means
Find prime factors of each number, identify common ones, use smallest power of each
LCM using prime factors: multiply all factors with highest powers
Mnemonic
Least Common Multiple = All factors with Highest powers (LCM = AHp)
When to use
When finding the smallest number that is divisible by two or more numbers
What each part means
Take all prime factors that appear, use the highest power of each factor
Quick recall chains
PEMDAS Order Steps
Recall test
What operation comes after exponents but before addition in PEMDAS?
Memory chain
Please Excuse My Dear Ate Sally - she always follows this order when cooking her famous adobo recipe
Items to remember
- Parentheses
- Exponents
- Multiplication/Division
- Addition/Subtraction
Divisibility Rules for 2,3,4,5,6,9,10
Recall test
Which divisibility rule requires checking if a number is both even and has digits that sum to a multiple of 3?
Memory chain
Even Three-year-old Four-wheelers Stop at Six Nine-story buildings - a story connecting each rule's key feature
Items to remember
- 2:even last digit
- 3:digit sum ÷ 3
- 4:last 2 digits ÷ 4
- 5:ends in 0 or 5
- 6:even AND ÷ 3
- 9:digit sum ÷ 9
- 10:ends in 0
Integer Operation Sign Rules
Recall test
What is the result when you multiply or divide two numbers with different signs?
Memory chain
Like attracts Like (same = positive), Opposites fight (different = negative), Works for both Multiply and Divide
Items to remember
- Same signs = positive
- Different signs = negative
- This applies to both multiplication and division
Steps for Prime Factorization
Recall test
After dividing repeatedly by 2, what should you do next in prime factorization?
Memory chain
Small prime Starts, Divide repeatedly, Move to Next when stuck, Continue until One, Write the Product
Items to remember
- Start with smallest prime (2)
- Divide repeatedly by same prime
- Move to next prime when no longer divisible
- Continue until quotient is 1
- Write as product of primes
Scientific Notation Conversion Steps
Recall test
If you move the decimal point 3 places to the right, what sign should the exponent have?
Memory chain
Place decimal, Count moves, Left = Positive, Right = Negative, Write the form
Items to remember
- Place decimal after first non-zero digit
- Count places moved
- Left moves = positive exponent
- Right moves = negative exponent
- Write in a × 10^n form
Diagrams
Integers, PEMDAS & Divisibility Concept Map
A comprehensive mind map showing the relationships between integers, order of operations, and divisibility concepts
PEMDAS Order of Operations Process
A flowchart showing the step-by-step process for applying the order of operations
Divisibility Testing Process
A decision tree for systematically testing divisibility rules
Integer Sign States in Operations
State diagram showing how integer signs change through different operations