Section 2.4: Divisibility and Prime Numbers in Word Problems
Key Takeaways
- Prime numbers have exactly two factors: 1 and themselves (note that 2 is the only even prime number).
- Divisibility rules are time-saving shortcuts that help bypass manual long division.
- Use Greatest Common Divisor (GCD) for problems involving splitting or dividing items into equal maximum groups.
- Use Least Common Multiple (LCM) for problems involving repeating cycles or events synchronizing in the future.
Section 2.4: Divisibility and Prime Numbers in Word Problems
Questions involving number theory concepts—such as divisibility, prime numbers, Greatest Common Divisor (GCD), and Least Common Multiple (LCM)—are stealthy. They often appear as complex word problems about scheduling, cutting materials, or arranging items, masking the underlying arithmetic requirements.
1. Prime Numbers and Prime Factorization
A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Note that 2 is the only even prime number.
A composite number has more than two factors. The number 1 is neither prime nor composite.
Prime Factorization is the process of breaking down a composite number into a product of prime numbers. This is typically done using a "factor tree".
Worked Example 1: Find the prime factorization of 60.
- 60 = 6 × 10
- 6 = 2 × 3 (both prime)
- 10 = 2 × 5 (both prime)
- Therefore, the prime factorization of 60 is 2 × 2 × 3 × 5, or 2² × 3 × 5.
Prime factorization is the fundamental tool for efficiently finding LCMs and GCDs.
2. Divisibility Rules
Knowing divisibility rules will save you enormous amounts of time, preventing long division to check for factors. Memorize these shortcuts:
- Divisible by 2: The number is even (ends in 0, 2, 4, 6, or 8).
- Divisible by 3: The sum of the digits is divisible by 3 (e.g., 234 -> 2+3+4=9, 9 is divisible by 3).
- Divisible by 4: The last two digits form a number divisible by 4 (e.g., 1,524 -> 24 is divisible by 4).
- Divisible by 5: The number ends in 0 or 5.
- Divisible by 6: The number is divisible by BOTH 2 and 3.
- Divisible by 8: The last three digits form a number divisible by 8.
- Divisible by 9: The sum of the digits is divisible by 9.
- Divisible by 10: The number ends in 0.
3. Greatest Common Divisor (GCD)
The Greatest Common Divisor (also called the Greatest Common Factor) is the largest positive integer that divides two or more numbers without a remainder.
When to use GCD in word problems:
You need the GCD when you are splitting things into smaller sections, arranging items into rows or groups, or trying to find the "maximum" size or "largest" capacity. Keywords include: split, divide evenly, cut into equal pieces, largest, maximum, biggest.
How to find the GCD:
- Find the prime factorization of each number.
- Identify the prime factors they have in common.
- Multiply those common prime factors together (using the lowest power of each common factor).
Worked Example 2: A carpenter has two pieces of wood. One is 48 inches long and the other is 72 inches long. He wants to cut both pieces into smaller pieces of equal length with no wood leftover. What is the greatest possible length he can cut them into?
- Prime factorization of 48: 2⁴ × 3
- Prime factorization of 72: 2³ × 3²
- Common factors: 2 and 3.
- Lowest powers: 2³ and 3¹.
- GCD = 2³ × 3 = 8 × 3 = 24. The greatest possible length is 24 inches.
4. Least Common Multiple (LCM)
The Least Common Multiple is the smallest positive integer that is a multiple of two or more numbers.
When to use LCM in word problems:
You need the LCM when events are repeating and you want to know when they will happen at the same time again. Keywords include: meet again, simultaneous, next time, cycle, together, least, minimum, smallest.
How to find the LCM:
- Find the prime factorization of each number.
- Write down all the prime factors that appear in any of the numbers.
- Use the highest power of each prime factor.
- Multiply them together.
Worked Example 3: Three different alarms are set to ring. The first rings every 12 minutes, the second every 15 minutes, and the third every 20 minutes. If they all ring together at 12:00 PM, at what time will they next ring together?
- Prime factorization of 12: 2² × 3
- Prime factorization of 15: 3 × 5
- Prime factorization of 20: 2² × 5
- List all factors used: 2, 3, 5.
- Highest powers: 2², 3¹, 5¹.
- LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60.
They will ring together again in 60 minutes. Therefore, the time will be 1:00 PM.
5. Remainders in Word Problems
Sometimes, you'll encounter division problems involving remainders. A classic problem type involves finding a number that leaves a specific remainder when divided by several numbers.
Worked Example 4: What is the smallest positive integer that leaves a remainder of 2 when divided by 4, 5, and 6?
Step 1: Find the LCM of the divisors (4, 5, and 6).
- 4 = 2²
- 5 = 5¹
- 6 = 2¹ × 3¹
- LCM = 2² × 3 × 5 = 60. The number 60 is perfectly divisible by 4, 5, and 6.
Step 2: Add the remainder. Since you want a remainder of 2, simply add 2 to the LCM. 60 + 2 = 62. The answer is 62.
Final Review of Strategic Tips
- Use the divisibility rules as a fast filter for multiple-choice options. If you know the answer must be a multiple of 9, instantly eliminate choices whose digits don't sum to a multiple of 9.
- The number 1 is not a prime number. The number 2 is the only even prime number. Do not forget these exceptions.
- Carefully analyze whether a problem is asking you to split things up (GCD) or synchronize repeating cycles (LCM).
A florist has 36 red roses and 54 white roses. She wants to create identical bouquets with no flowers left over. What is the maximum number of bouquets she can make?
Train A departs every 24 minutes and Train B departs every 30 minutes. If both trains depart simultaneously at 8:00 AM, at what time will they next depart together?
Which of the following numbers is divisible by both 4 and 9?