Cool numbers!

Number Theory Level 4

We call a number to be cool if it can be expressed as $25a+15b+9c$ for non-negative integers $a$ , $b$ , and $c.$ What is the largest integer which isn't cool?

The answer is 71.

1 solution

Ivander Jonathan
Feb 21, 2015

First, we need to factor some of the numbers. $\huge{5(5a+3b)+9c}$ The largest number that cannot be expressed as numbers in the brackets is $7$ .

So, the equation is $5\cdot7+9c$ to find the largest uncool number.

$c$ must not be 5 because it leads to $5(7+9)$ which will be a cool number because $7+9$ could be expressed as $5\cdot2+3\cdot2$ .

Hence $c=4$ to find the uncool number.

Substitution: $5\cdot7+9\cdot 4=\boxed{71}$

I don't understand, why can't you start with $25a + 3(5b+ 3c)$ ?

Shortcut: Knowing that 25,15,9 follows a geometric progression, we can immediately apply the formula from here to get 71.

Pi Han Goh - 5 years, 6 months ago

It does work. In this way, the largest number that the number in the brackets cannot express 7. While the variable $a$ must be less than $3$ since $3$ is the multiplier of the number in brackets. Then $a=2$ and $25\cdot2+3(7)=50+21=\boxed{71}$ .

Ivander Jonathan - 4 years, 9 months ago

Cool numbers!

The answer is 71.

1 solution

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