Absolute minimum

Algebra Level 2

If $n$ is an integer, then what is the least possible value of

$\left| 123 - 5n \right| ?$

0 1 2 5 3 4

2 solutions

Vaibhav Prasad
Feb 20, 2015

The multiple of $5$ closest to $123$ is $125$ and

$\left| 123-125 \right|$ = 2

Any multiple of $5$ less than or greater than $125$ will give a larger value than $2.$ The next best is $|120-123| = 3.$

Well done. I think that a lot of people chose 3 as the answer.

Chung Kevin - 6 years, 3 months ago

The question is incomplete. The smallest value of ABS(123-5*n) occurs when N=5. It is 98. The expression has a value of 113 when n=2.

CHARLES GALLAGHER - 5 years, 5 months ago

I also agree with you .The answer must be 5

Nguyễn Hưng - 5 years, 4 months ago

The question asked is to find the least possible value of abs(123-5n) for any given integer n and not to find the value of n for which abs(125-5n) is least.

Shanthanu Rai - 5 years, 3 months ago

yeah.. i wonder how people get 2 as a answer???

Tanmay Kaviwala - 5 years, 5 months ago

Right, because 123-10 = 2.

Sara Flint - 5 years, 3 months ago

The trick is in the wording. We aren't trying to find the value of n, we're asking to find the lowest possible outcome of Abs(123-5n) if n is an integer. Therefore if n=25, then the value is 2. If n=24, then value is 3.

Jay Carter - 5 years ago

Magnus Frennberg
Jun 10, 2018

The questions is, if n is any integer, which is least possible absolute value of $|123 - 5n|$ , as in the value closest to 0. We are not looking for the lowerst possible value of $n$ .

Therefor, we solve this by looking at how close we can get to 0 by making $n$ any interger of our choice. Using modulus, we get that $\frac{123}{5} \equiv 24 + 3$ where 3 is the remainder after dividing 123 by 24.

You might think that you are done now, and that the answer is 3, but since we are working in absolute numbers we can get even closer to 0 by going into negative numbers.

Therefore, we add 1 to our n making it 25.

So the answer is:

$|123 - (25 \cdot 5) | = | 123 - 125 | = |-2| = 2$

And that is why the correct answer, the closest we can get to zero following the premises, is 2.

Bonus : Also, a more "mathematical" reason to add 1 to n is because our remainder is larger than x divided by 2. We can replace 5 with any positive integer and get this rule:

if $\frac{x}{2} > remainder$ then add 1 to n.

Absolute minimum

2 solutions

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