Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

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What Does This Problem Mean?

Understanding the Problem

Problem Statement: "Find the greatest number that will divide 43, 91, and 183 so as to leave the same remainder in each case."

What this means: We're looking for a number that:

• Divides 43 and leaves some remainder (let's say remainder = r)
• Divides 91 and leaves the same remainder r
• Divides 183 and leaves the same remainder r

And this number should be the greatest (largest) such number.

Visual Example

Let's say we find that the number is 4:

• 43 ÷ 4 = 10 remainder 3
• 91 ÷ 4 = 22 remainder 3
• 183 ÷ 4 = 45 remainder 3

Notice: All three divisions leave the same remainder (3). And 4 is the greatest number that does this.

Real-World Application

This concept is useful in:

• Distribution problems: Dividing items into groups where each group has leftover items
• Time scheduling: Finding cycles that align differently occurring events
• Inventory management: Organizing products where some items always remain

Why Does This Method Work? The Mathematical Logic

The Key Insight

When a number divides two different numbers and leaves the same remainder, it must divide their difference exactly (with no remainder).

Proof Through Example

Let's say number D divides both 91 and 183 leaving remainder 3:

• 91 = D × q₁ + 3 (where q₁ is quotient)
• 183 = D × q₂ + 3 (where q₂ is quotient)

Subtracting the first equation from the second:

• 183 − 91 = D × q₂ + 3 − (D × q₁ + 3)
• 92 = D × (q₂ − q₁) + 0
• 92 = D × (q₂ − q₁)

This means D divides 92 exactly (no remainder)!

Generalization

If a number D divides three numbers (43, 91, 183) with the same remainder, then:

• D divides (183 − 91) = 92
• D divides (183 − 43) = 140
• D divides (91 − 43) = 48

Therefore, D must be a common divisor of 92, 140, and 48.

The greatest such D is the HCF of these differences.

💡 Why This Works: The remainder "cancels out" when we subtract. Only the divisor part remains. So finding HCF of differences gives us the greatest divisor that produces equal remainders.

Step-by-Step Method to Solve

The Complete Algorithm

Step 1: Write down all numbers

• Original numbers: a, b, c

Step 2: Find all pairwise differences

• Difference 1: |b − a|
• Difference 2: |c − a|
• Difference 3: |c − b|
• (Or just pick two differences; the HCF will be the same)

Step 3: Find prime factorization of each difference

• Factor each difference completely

Step 4: Identify common factors

• Find factors that appear in ALL differences

Step 5: Multiply common factors to get HCF

• The product of all common factors = HCF = Answer

Quick Reference Chart

Solved Example 1: Find Greatest Number Dividing 43, 91, 183

Complete Solution (As Given)

Given Numbers: 43, 91, 183

Step 1: Find Pairwise Differences

$$183-91=92$$ $$183-43=140$$ $$91-43=48$$

Step 2: Prime Factorization of Each Difference

92=22×2392 = 2^2 \times 2392=22×23 140=22×5×7140 = 2^2 \times 5 \times 7140=22×5×7 48=24×348 = 2^4 \times 348=24×3

Step 3: Identify Common Factors

Looking at the prime factorizations:

• 92 has: 2², 23
• 140 has: 2², 5, 7
• 48 has: 2⁴, 3

Common factor present in ALL three: 2² = 4

Step 4: Calculate HCF

HCF(92,140,48)=22=4\text{HCF}(92, 140, 48) = 2^2 = 4HCF(92,140,48)=22=4

Answer: 4

Verification

Let's verify that 4 is correct:

• 43 ÷ 4 = 10 remainder 3 ✓
• 91 ÷ 4 = 22 remainder 3 ✓
• 183 ÷ 4 = 45 remainder 3 ✓

All three divisions leave the same remainder (3). ✓

Solved Example 2: Different Numbers, Same Method

Problem

Find the greatest number that divides 65, 117, and 169 leaving the same remainder.

Solution

Step 1: Find Pairwise Differences

$$169-117=52$$ $$169-65=104$$ $$117-65=52$$

(Notice: 169−117 and 117−65 give the same difference. This happens sometimes!)

Step 2: Prime Factorization

52=22×1352 = 2^2 \times 1352=22×13 104=23×13104 = 2^3 \times 13104=23×13

(We only need two different differences for HCF)

Step 3: Common Factors

• 52 has: 2², 13
• 104 has: 2³, 13

Common factors: 2² and 13

Step 4: Calculate HCF

HCF(52,104)=22×13=4×13=52\text{HCF}(52, 104) = 2^2 \times 13 = 4 \times 13 = 52HCF(52,104)=22×13=4×13=52

Answer: 52

Verification

• 65 ÷ 52 = 1 remainder 13 ✓
• 117 ÷ 52 = 2 remainder 13 ✓
• 169 ÷ 52 = 3 remainder 13 ✓

All leave remainder 13. The answer is correct!

How to Find HCF Using Prime Factorization

Method Explained

HCF (Highest Common Factor) is found by:

1. Writing each number as a product of prime factors
2. Identifying which prime factors appear in ALL numbers
3. Multiplying the common factors (taking lowest power if they repeat)

Example with the Original Problem

Numbers: 92, 140, 48

Common factors (in ALL three): 2²

HCF = 2² = 4 ✓

Why Prime Factorization Works

Prime factorization breaks numbers into their simplest building blocks (prime numbers). Common building blocks are the factors shared by all numbers. Multiplying these shared blocks gives the HCF.

Alternative: Finding HCF Using Euclidean Algorithm

What is the Euclidean Algorithm?

The Euclidean Algorithm is a method to find HCF using repeated division:

• Divide larger number by smaller
• Replace larger with smaller, smaller with remainder
• Repeat until remainder is 0
• The last non-zero remainder is the HCF

Step-by-Step Example

Find HCF(140, 92):

$$140=92\times 1+48$$ $$92=48\times 1+44$$ $$48=44\times 1+4$$ $$44=4\times 11+0$$

The HCF is 4 (last non-zero remainder)

Advantages of Euclidean Algorithm

For exams: Use prime factorization if numbers are small (under 1000). Use Euclidean Algorithm if numbers are large.

Common Mistakes Students Make

Mistake 1: Forgetting to Find Differences First

Wrong Approach: Finding HCF of original numbers directly

• HCF(43, 91, 183) = 1 ❌

Correct Approach: Find HCF of differences

• HCF(92, 140, 48) = 4 ✓

Why the mistake happens: Students think "find HCF of given numbers" without understanding why differences are needed.

Mistake 2: Finding Only Some Differences

Wrong Approach:

• Difference 1: 183 − 91 = 92
• Finding HCF of just 92 ❌

Correct Approach: Find at least 2–3 differences and then HCF

• Differences: 92, 140, 48
• HCF(92, 140, 48) = 4 ✓

Why this matters: Using only one difference might give wrong answer if that difference has extra factors.

Mistake 3: Misunderstanding "Same Remainder"

Common Confusion: "Same remainder" means remainder = 0

Correct Understanding: "Same remainder" means all divisions leave the same remainder (could be 0, could be 3, could be any value)

Example: The answer 4 leaves remainder 3 in all cases—not remainder 0!

Mistake 4: Calculating HCF Incorrectly

Wrong Approach: Multiplying all common prime factors without checking

92 = 2² × 23 140 = 2² × 5 × 7 48 = 2⁴ × 3

Mistakenly thinking: Common factors are 2, 2, 3, so HCF = 2 × 2 × 3 = 12 ❌

Correct Approach: Find prime factors present in ALL three

• Only 2² is in all three
• HCF = 2² = 4 ✓