Finding the Greatest Divisor with Specific Remainders

The question asks for the largest number (let's denote it by N) that divides 13, 26, and 39, and leaves specific remainders: 1, 2, and 3, respectively.

Step 1: Determine the Divisible Numbers

If a number N divides a number A leaving a remainder R, then N must divide exactly into the difference $A - R$. We apply this to each case:

• For the number 13 with remainder 1: The number to be divided exactly is $13 - 1 = 12$.
• For the number 26 with remainder 2: The number to be divided exactly is $26 - 2 = 24$.
• For the number 39 with remainder 3: The number to be divided exactly is $39 - 3 = 36$.

So, the required number N must be a common divisor of 12, 24, and 36.

Step 2: Calculate the Highest Common Factor (HCF)

Since we need the *greatest* number that divides 12, 24, and 36, we need to find the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of these three numbers.

We list the factors of each number:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The common factors are 1, 2, 3, 4, 6, and 12. The greatest among these is 12.

Alternatively, using prime factorization:

• $12 = 2^2 \times 3^1$
• $24 = 2^3 \times 3^1$
• $36 = 2^2 \times 3^2$
• The HCF is found by taking the lowest power of each common prime factor: $HCF(12, 24, 36) = 2^{\min(2,3,2)} \times 3^{\min(1,1,2)} = 2^2 \times 3^1 = 4 \times 3 = 12$.

Step 3: Verify the Result

We check if 12 divides the original numbers with the specified remainders:

• $13 \div 12$: $13 = 1 \times 12 + 1$. Remainder is 1. Correct.
• $26 \div 12$: $26 = 2 \times 12 + 2$. Remainder is 2. Correct.
• $39 \div 12$: $39 = 3 \times 12 + 3$. Remainder is 3. Correct.

The number 12 meets all the conditions.

Final Answer: The greatest number is 12.