Integers, PEMDAS & Divisibility - Flashcards
Master the fundamental concepts of integers, order of operations (PEMDAS), and divisibility rules with these comprehensive flashcards. These cards cover essential numerical ability concepts tested in major Philippine exams including UPCAT, CSE, LET, NLE, NMAT, ACET, and USTET. Each card reinforces understanding through clear examples and practical applications.
Flashcards
What are integers and how are they classified?
Answer
Integers are whole numbers that can be positive, negative, or zero, but cannot be fractions. They are represented by the symbol Z. Classification: (1) Zero - neither positive nor negative, (2) Positive integers - natural numbers like 1, 2, 3..., (3) Negative integers - like -1, -2, -3... Zero is an even number and is neither prime nor composite.
What does PEMDAS stand for and what is the correct order of operations?
Answer
PEMDAS stands for: P - Parentheses, E - Exponentiation, M - Multiplication, D - Division, A - Addition, S - Subtraction. Order: (1) Parentheses first (innermost outward), (2) Exponents, (3) Multiplication and Division (left to right), (4) Addition and Subtraction (left to right). Remember: MD and AS have equal priority and are done left to right.
Add the following integers: (-15) + 23
Answer
(-15) + 23 = 8. When adding integers with different signs, subtract their absolute values (23 - 15 = 8) and use the sign of the number with the larger absolute value. Since 23 > 15 and 23 is positive, the result is positive.
Solve: 2 - [4 - 3 × (-3 + 5)²] + 4 ÷ 2 × 6 - 3
Answer
Following PEMDAS: Step 1: Parentheses (-3 + 5)² = (2)² = 4. Step 2: Brackets [4 - 3 × 4] = [4 - 12] = -8. Step 3: Division/Multiplication 4 ÷ 2 × 6 = 2 × 6 = 12. Step 4: Final calculation 2 - (-8) + 12 - 3 = 2 + 8 + 12 - 3 = 19.
State the divisibility rule for 3 and provide an example.
Answer
A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 423 is divisible by 3 because 4 + 2 + 3 = 9, and 9 ÷ 3 = 3. Therefore, 423 ÷ 3 = 141 with no remainder.
What are the special properties of zero in arithmetic operations?
Answer
Zero properties: (1) Adding/subtracting zero: n + 0 = n, n - 0 = n. (2) Multiplying by zero: n × 0 = 0. (3) Dividing zero: 0 ÷ n = 0 (if n ≠ 0). (4) Dividing by zero: n ÷ 0 is undefined. (5) Zero to any power: 0ⁿ = 0. (6) Any number to the power of zero: n⁰ = 1 (except 0⁰ which is indeterminate).
Multiply the following integers: (-7) × (-12)
Answer
(-7) × (-12) = 84. When multiplying two integers with the same sign (both negative), the result is positive. The value is the product of their absolute values: 7 × 12 = 84.
State the divisibility rules for 2, 5, and 10.
Answer
Divisibility by 2: Number must be even (last digit is 0, 2, 4, 6, or 8). Divisibility by 5: Last digit must be 0 or 5. Divisibility by 10: Last digit must be 0. Examples: 124 (÷2), 405 (÷5), 640 (÷10).
Find the absolute value: |-15| and |8|
Answer
|-15| = 15 and |8| = 8. The absolute value of a number is its distance from zero, always positive. For negative numbers, remove the negative sign. For positive numbers and zero, the absolute value equals the number itself.
Subtract: 18 - (-4)
Answer
18 - (-4) = 18 + 4 = 22. To subtract a negative number, change the operation to addition and change the sign of the second number. Subtracting a negative is the same as adding a positive.
Is 312 divisible by 6? Explain your reasoning.
Answer
Yes, 312 is divisible by 6. A number is divisible by 6 if it's divisible by both 2 and 3. Check: (1) 312 is even (last digit 2), so divisible by 2. (2) Sum of digits: 3 + 1 + 2 = 6, and 6 ÷ 3 = 2, so divisible by 3. Since both conditions are met, 312 ÷ 6 = 52.
Evaluate: 5 × (13 - 2) + 45
Answer
Following PEMDAS: Step 1: Parentheses first (13 - 2) = 11. Step 2: Multiplication 5 × 11 = 55. Step 3: Addition 55 + 45 = 100. Answer: 100.
What is the law of signs for multiplication and division of integers?
Answer
Law of Signs: (1) Same signs → Positive result (+ × + = +, - × - = +, + ÷ + = +, - ÷ - = +). (2) Different signs → Negative result (+ × - = -, - × + = -, + ÷ - = -, - ÷ + = -). Remember: 'Same signs give positive, different signs give negative.'
State the divisibility rule for 9 and test if 6957 is divisible by 9.
Answer
Rule: A number is divisible by 9 if the sum of its digits is divisible by 9. Test 6957: 6 + 9 + 5 + 7 = 27. Since 27 ÷ 9 = 3, the number 6957 is divisible by 9. Verification: 6957 ÷ 9 = 773.
Solve: (10 - 6)² + 16 ÷ 4 × 6 - 14 + 2
Answer
Following PEMDAS: Step 1: Parentheses (10 - 6) = 4. Step 2: Exponent 4² = 16. Step 3: Division/Multiplication left to right: 16 ÷ 4 = 4, then 4 × 6 = 24. Step 4: Addition/Subtraction left to right: 16 + 24 - 14 + 2 = 40 - 14 + 2 = 26 + 2 = 28.
What are prime and composite numbers? Give examples.
Answer
Prime numbers: Greater than 1, divisible only by 1 and themselves. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23. Composite numbers: Have more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15. Note: 1 is neither prime nor composite. 2 is the only even prime number.
Test if 2944 is divisible by 8 using the divisibility rule.
Answer
Rule for 8: A number is divisible by 8 if its last three digits form a number divisible by 8. Test 2944: Last three digits are 944. Check: 944 ÷ 8 = 118 (exact division). Therefore, 2944 is divisible by 8. Verification: 2944 ÷ 8 = 368.
Add three integers: (-8) + (-12) + 15
Answer
Step 1: Add the negative integers first: (-8) + (-12) = -20. Step 2: Add the positive integer: (-20) + 15 = -5. When adding integers with different signs, subtract absolute values and use the sign of the larger absolute value. Since |-20| > |15|, result is negative.
What is the divisibility rule for 4? Test if 536 is divisible by 4.
Answer
Rule for 4: A number is divisible by 4 if its last two digits form a number divisible by 4, or if it ends in 00. Test 536: Last two digits are 36. Check: 36 ÷ 4 = 9 (exact division). Therefore, 536 is divisible by 4. Verification: 536 ÷ 4 = 134.
Compare the expressions: 3² × 4 and (3 × 4)². Which is larger?
Answer
Expression 1: 3² × 4 = 9 × 4 = 36. Expression 2: (3 × 4)² = 12² = 144. Therefore, (3 × 4)² = 144 is larger than 3² × 4 = 36. This demonstrates the importance of parentheses in changing the order of operations and significantly affecting the result.
Explain the divisibility rule for 11 and test if 4510 is divisible by 11.
Answer
Rule for 11: A number is divisible by 11 if the alternating sum of its digits (or difference between sum of digits in odd positions and even positions) is 0 or divisible by 11. Test 4510: Odd positions (1st, 3rd): 4 + 1 = 5. Even positions (2nd, 4th): 5 + 0 = 5. Difference: 5 - 5 = 0. Since 0 is divisible by 11, 4510 is divisible by 11.
What happens when you divide integers with different signs? Give an example.
Answer
When dividing integers with different signs, the quotient is always negative. The value is the quotient of their absolute values. Examples: (+15) ÷ (-3) = -5, (-20) ÷ (+4) = -5. Rule: Different signs in division always produce a negative result.
Diagrams
Integers, PEMDAS & Divisibility Overview
A comprehensive mind map showing the relationship between integers, PEMDAS order of operations, and divisibility rules, providing students with a visual overview of all key concepts in numerical ability.
PEMDAS Order of Operations Process
A step-by-step flowchart showing how to apply the PEMDAS rule when solving mathematical expressions, helping students follow the correct order of operations systematically.