# Find all triples $(a,b,c)$ of positive integers such that $(1+\frac{1}{a})(1+\frac{1}{b})(1+\frac{1}{c})=3$ [duplicate]

Find all triples $(a,b,c)$ of positive integers such that $(1+\frac{1}{a})(1+\frac{1}{b})(1+\frac{1}{c})=3$

Supposing $a\ge b\ge c$, $$(1+\frac{1}{c})^3\ge 3$$ $$\Rightarrow 1 + \frac{1}{c^3} + 3(1 + \frac{1}{c})\frac{1}{c} \ge3$$ After this how to proceed?

## marked as duplicate by rlartiga, MAN-MADE, Martin R, Thomas Andrews elementary-number-theory StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Sep 13 '17 at 17:53

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• starting with $a\geq b\geq c$ is a good idea, but in this way you can not get triples, I guess – MAN-MADE Sep 13 '17 at 17:12
• Since $(1+ \frac{1}{3})^3 < 3$, you must have $c\le 2$. – Orest Bucicovschi Sep 13 '17 at 17:18