If the denominator of a rational number contains only 2 and 5 as prime factors then the decimal expansion of the rational number is terminating.  How can I Prove this
Show that a number has a terminating decimal expansion if and only if, it is rational and when in lowest terms, its denominator is coprime to all primes other than $2$ and $5$.   This is an unsolved question in my lecture notes. I can only seem to prove the converse direction for this. Would appreciate a solution for the other direction.     For converse direction:   Let the number, in lowest terms, be given by $\frac{l}{m}$ where $m = 2^ \alpha 5^\beta$, for some positive integers, $\alpha$ and $\beta$.    If $\alpha &gt; \beta$, let $k = 5^{(\alpha-\beta)}$. If $\alpha &lt; \beta$, let $k = 2^{(\beta-\alpha)}$.   Then $\frac{l}{m} = \frac{kl}{k2^\alpha5^\beta}=\frac{kl}{10^q}$ where $q = \max(\alpha,\beta)$.   Hence $\frac{l}{m}$ is a terminating decimal.
Simple Hadamard Circuit gives incorrect results? How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$? Maximum and minimum distance of two points Consider six distinct points in a plane. Let $m$ and $M$ denote respectively the minimum and the maximum distance between any pair of points.  Show that $M/m \geqslant \sqrt{3}$. Is the ideal $(2,X+1)\subset\Bbb{Z}[X]$ prime, maximal or neither? I'm currently considering the ideal $(2,X+1)\subset\Bbb{Z}[X]$. I'm trying to figure out if this is a prime ideal, a maximal ideal, or neither. So far I have attempted to look at the quotient ring $\Bbb{Z}[X]/(2,X+1)$. I know that if this quotient ring is a field, then our ideal is maximal, since $\Bbb{Z}[X]$ is commutative. However I'm stuck on how I should be viewing this quotient ring. 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The fact that we can't precisely know both the position and momentum of an elementary particle is, indeed, a property of the particle itself. It is an intrinsic property of the Universe we live in.   Then this video came out:  (skip to , if you're already familiar with the uncertainty principle).  So, correct me if I'm wrong, what we may claim according to the video is:     the only way to measure an elementary particle is to make it interact with another elementary particle: it is therefore incorrect to say that an elementary particle doesn't have a well defined momentum/position before we make our measurement. We cannot access this data (momentum/position) without changing it, therefore it is correct to say that our ignorance about this data is not an intrinsic property of the Universe (but, rather, an important limit of how we can measure it).   Please tell me how can both of the highlighted paragraphs be true or how they should be corrected. 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