Question

Find the greatest number by which when 472, 832 and 1372 are divided, the remainder are the same.

Answer 1

Hint: In this question, we need to find the greatest number by which all the three given numbers, when divided, leave the same remainder. So, we first find the differences between any two pairs of the given numbers and then find the greatest common divisor of the differences to get to the required answer. Greatest common divisor is the greatest number by which both the numbers are divisible.

Complete step by step answer:
We need to find the greatest number by which when 472, 832 and 1372 are divided, the remainder are the same. For this we first find the difference of the given numbers 472, 832 and 1372.The differences are
$832-472=360 \\
\Rightarrow 1372-832=540 \\ $
Now we find the GCD of the differences.We use the simultaneous factorisation to find the greatest common factor of 360 and 540.We have to divide both of them with possible primes which can divide both of them.
\[\begin{align}
  & 2\left| \!{\underline {\,
  360,540 \,}} \right. \\
 & 2\left| \!{\underline {\,
  180,270 \,}} \right. \\
 & 3\left| \!{\underline {\,
  90,135 \,}} \right. \\
 & 3\left| \!{\underline {\,
  30,45 \,}} \right. \\
 & 5\left| \!{\underline {\,
  10,15 \,}} \right. \\
 & 1\left| \!{\underline {\,
  2,3 \,}} \right. \\
\end{align}\]

The GCD is $2\times 2\times 3\times 3\times 5=180$.

Therefore, the greatest number by which when 472, 832 and 1372 are divided, the remainder are the same is 180.

Note: We need to remember that the GCF has to be only one number. It is the greatest possible divisor of all the given numbers. If the given numbers are prime numbers, then the GCD of those numbers will always be 1. Therefore, if for numbers $x$ and $y$, the GCD is $a$ then the GCD of the numbers $\dfrac{x}{a}$ and $\dfrac{y}{a}$ will be 1.