Question

What is the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively?

Answer 1

Hint: For solving this question you should know about the Greatest Common Factor between the two or more digits. For calculating the greatest common factor or GCF, we will do factors of the given digits and then we will take all the common values and see the greatest common value in that. But before that, we will subtract the remainders of that and then we find some values and these values will give us the greatest common factor or the largest number which will divide them.

Complete step-by-step solution:
According to our question we have to calculate the largest number which divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively. We can solve this by the factorisation method. IN this method we make factors of all the digits which are given and then take the common digit outside and then we see the greatest common value in them.
If we see an example: Find the GCF of 18, 27.
Then the factors of 18 = 1, 2, 3, 6, 9, 18
And the factors of 27 = 1, 3, 9, 27
The common factors of 18 and 27 are 1, 3 and 9. Among these numbers, 9 is the greatest (Larger) Thus the GCF of 18 and 27 is 9.
So this can be written as: GCF (18, 27) = 9.
OR we can solve this by division method also.
Step 1: Divide the larger number by the smaller number to give a quotient and remainder.
$\Rightarrow \dfrac{27}{18}=1$ with a remainder of 9.
Step 2: If the remainder is zero, then the smaller number is GCF. But here it is not.
Step 3: Repeat with the smaller number and remainder this step again.
$\Rightarrow \dfrac{18}{9}=2$ with a remainder 0.
Here the remainder is zero, So, GCF (27, 18) = 9.
But here as our question is asked, then:
$\begin{align}
  & \Rightarrow 626-1=625 \\
 & \Rightarrow 3127-2=3125 \\
 & \Rightarrow 15628-3=15625 \\
\end{align}$
So, now the new numbers are: 625, 3125, 15625.
Now using Euclid’s division lemma to find the HCF of 625, 3125 and 15625:
So, HCF of 625 and 3125:
$3125=625\times 5+0$
Clearly the HCF of 625 and 3125 is 625.
Now the HCF of 625 and 15625:
$15625=625\times 25+0$
So, the HCF of 15625 and 625 is 625.
Hence HCF of 625, 3125 and 15625 is 625.

Note: During solving this question you can use any one of the methods, but the first method is more tough and that has chances of mistakes in the answer because factors can be wrong and by mistake, we can forget to take the common one from the value of the factors, in which case our answer would be wrong.