Question

Find the largest no which divides 70 and 125 leaving remainder 5 and 8 respectively.

Answer 1

Hint: We’ll use the formula dividend = divisor \[ \times \] quotient + remainder and subtract remainders from the given numbers respectively, then we find the HCF of those two, that will give us the required number.

Complete step by step solution: We have been given numbers 70 and 125 which leaves remainder 5 and 8 respectively when divides by a number.
Subtracting the $1^{\text{st}}$ remainder from the $1^{\text{st}}$ number \[ = (70 - 5) = 65\]
Subtracting the $2^{\text{st}}$ remainder from the $2^{\text{st}}$ number \[ = (125 - 8) = 117\]
These are divisible by the required number.

Now, required number = HCF of 65, 117 (for the largest number)
For this, we apply the long division method to get the HCF,
Where 117 is dividend and 65 is divisor,
 \[117 = 65 \times 1 + 52\] [dividend = divisor \[ \times \] quotient + remainder]
Now we have 65 as dividend and 52 as divisor,
\[ \Rightarrow \]65 = 52 \[ \times \]1 + 13
Now we have 52 as dividend and 13 as divisor,
\[ \Rightarrow \]52 = 13 \[ \times \]4 + 0
We stop here as the remainder is zero,
\[\therefore \] so we have, HCF = 13

Hence, 13 is the largest number which divides 70 and 125, leaving remainders 5 and 8.

Note: Need to keep that in mind that the HCF will bring out the largest number. We can also find the HCF by factorizing the numbers and getting the product of all the common factors.