Question 822570
I think the solution will be easier to understand if we rewrite the decimals as fractions:
{{{0.06 = 6/100 = 3/50}}}
{{{0.008 = 8/1000 = 1/125}}}
{{{0.16 = 16/100 = 4/25}}}<br>
Let x be the desired number. Then<ul><li>x divided by 3/50 is an integer.</li><li>x divided by 1/125 is an integer.</li><li>x divided by 4/25 is an integer.</li></ul>Since dividing by a fraction is done by multiplying by the fraction's reciprocal:<ul><li>x times 50/3 is an integer.</li><li>x times 125/1 is an integer.</li><li>x times 25/4 is an integer.</li></ul>If all these multiplications are integers, then the factors of each denominator <u>must</u> be canceled by factors of x. So let's look at the factors of the denominators (of the reciprocals):
3 = 3 * 1
1 = 1 * 1
4 = 2 * 2<br>
Since 1 is a factor of everything, x must only have one 3 and two 2's as factors in order for it to cancel the factors of each of the denominators. (any additional factors will just make x larger.) So:
x = 3 * 2 * 2 = 12