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Greatest Common Divisor, or 'gcd'.
gcd(900,1026):
Imagine you want to make two buildings 900' tall and 1026' tall.
What is the tallest measuring-stick which can measure
both buildings, *and* not have any fractional part left over?
For example, a three-foot rod would evenly measure both 900' and 1026'
(300x and 342x, resp).
But you know, a 6' rod would work too (150x and 171x, resp.).
But is 6' the *tallest* such rod? (It turns out: No!)
(Or, instead of measuring rods, suppose you are using brick-heights:
You want to use the same height brick on both buildings (to save cost),
and you want to use as tall a brick as possible (to save work).)
Well, *whatever* size measure rod we use, if it evenly divides 900 and 1026,
it must also evenly divide the difference, 1026-900=126.
In fact, gcd(1026,900)=gcd(90,36).
(That's true of *any* divisor, not just the gcd)
gcd(1026,900) We can subtract off a copy of the smaller building,
= gcd(900,1026-900) and the gcd must still divide the difference evenly.(*)
= gcd(900,126) (do the
= gcd(126, 900-7*126) Take off 7 copies of of the smaller building
= gcd(18,
(*) It's clear that any divisor of 1026 and 900 must be a divisor of their
difference; showing that the *greatest* common divisor of each is the
same takes a smidgen of work. (Try a proof-by-contradiction.)
Think of an infinite family of statements,
parameterized by a natural number.
That is, let P be a function:
P: N → formula
Induction will be a rule-of-inference: If we know
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| ©2008, Ian Barland, Radford University Last modified 2008.Dec.12 (Fri) |
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