HCF Notion #1

Number Theory Level 2

$\large \color{#20A900}{\boxed{3827} \quad \boxed{4171} \quad \boxed{3569} \quad \boxed{3053}}$

Find the highest common factor (HCF) of all the above boxed numbers.

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The answer is 43.

2 solutions

Michael Fuller
May 28, 2015

$3827=43\times 89$

$4171=43\times 97$

$3569=43\times 83$

$3053=43\times 71$

From this we can tell that $HCF\left( 3827,4171,3569,3053 \right) =\boxed { 43 }$ .

Moderator note:

Right. But how did you determine that $43$ divides all these four numbers? That's the most important part of this question.

Like Satvik Choudhary had mentioned, the best way (or simpler way) to is to find the pairwise difference between each numbers.

Below shows a couple of pairwise differences.

$|4171 - 3827| = 344 , |4171 - 3569| = 602 , |3827 - 3569| = 258$ .

It's easier to find the factors of smaller numbers. Now it's easily seen that $344 = 8 \times 43, 602 = 43 \times 14, 258 = 43 \times 6$ .

This suggests that $43$ is a highest common factor. Note that I used the word "suggests" because it haven't been verified yet.

Dividing all the four numbers gives $89, 97,83,71$ . Because at least one of them is a prime number (actually, all four of them are prime) and therefore can't have any more common factors, we can conclude that the HCF of these numbers is indeed $43$ .

LOL , the note is bigger than the solution. :P

Nihar Mahajan - 6 years ago

To Challenger Master. Why does the factors of those differences suggest the highest common factor of the four numbers?

Patrick Engelmann - 6 years ago

Satvik Choudhary
May 27, 2015

A good idea would be to try to find the common factors of the difference between these numbers.

HCF Notion #1

Join the Brilliant Classes and enjoy the excellence. Also checkout Foundation Assignment #2 for JEE.

The answer is 43.

2 solutions

Moderator note:

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