find the largest number that divides 125 and 183 leaving remainder 5 and 3 respectively

Answer: 60

Step-by-step solution

Step 1: Subtract the remainders from given numbers

If a number divides $125$ leaving a remainder of $5$ , it must exactly divide $125 - 5 = 120$ . Similarly, if it divides $183$ leaving a remainder of $3$ , it must exactly divide $183 - 3 = 180$ . Therefore, the required largest number is the highest common factor (HCF) of $120$ and $180$ .

Step 2: Find the prime factorisation of 120 and 180

We express both $120$ and $180$ as products of their prime factors using the Fundamental Theorem of Arithmetic. For $120$ , we have $120 = 2 \times 2 \times 2 \times 3 \times 5 = 2^3 \times 3^1 \times 5^1$ . For $180$ , we have $180 = 2 \times 2 \times 3 \times 3 \times 5 = 2^2 \times 3^2 \times 5^1$ .

Step 3: Compute the HCF of 120 and 180

The $\text{HCF}$ of two numbers is the product of the smallest power of each common prime factor involved in the numbers. The common prime factors are $2$ , $3$ , and $5$ with least powers $2^2$ , $3^1$ , and $5^1$ , giving $\text{HCF}(120, 180) = 4 \times 3 \times 5 = 60$ .

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