Question 5

The largest number which divides $70$ and $125$, leaving remainder $5\text{and}8$ respectively, is

Solution:

To find the largest number that divides 70 and 125, leaving remainders of 5 and 8 respectively, we can follow these steps:

Step 1: Adjust the numbers by subtracting the remainders
– For the first number (70), subtract the remainder (5):
$$70-5=65$$
– For the second number (125), subtract the remainder (8):
$$125-8=117$$

Step 2: Find the HCF (Highest Common Factor) of the adjusted numbers
We need to find the HCF of 65 and 117.

Step 3: Prime factorization of 65
– 65 can be factored as:
$$65=5\times 13$$

Step 4: Prime factorization of 117
– 117 can be factored as:
$$117=3\times 39$$
– Further factor 39:
$$39=3\times 13$$
– Thus, the complete factorization of 117 is:
$$117=3^2\times 13$$

Step 5: Identify the common factors
– The prime factors of 65 are $$5$$ and $$13$$.
– The prime factors of 117 are $$3^2$$ and $$13$$.
– The common factor is $$13$$.

Step 6: Conclusion
The largest number that divides both 70 and 125, leaving remainders of 5 and 8 respectively, is:
$$\text{HCF}=13$$

Final Answer
The largest number is 13.