A million days from now

February 21st, 2004 admin

I am reading a book called In Code - A young woman’s mathematical journey written by Sarah Flannery. In one of the chapters while trying to explain the concept of modulo arithmetic, she poses an interesting question along the lines of,
“Say today is Saturday, what day of the week will the millionth day from today be?”
The answer uses some very simple principles of modulo arithmetic. And I thought it was fascinating the way it is solved so easily. The question is simply,
If the days of the week are labeled 1,2…7 (for Monday, Tuesday, …, Sunday respectively) then the n^th day from today would be more than today by remainder when you divide n by 7.
n^th day from today = today + remainder when n is divided by 7 (or divided into 7, whichever convention you are comfortable with)
n^th day from today = today + n mod 7 (the mod part is therefore just another way of saying remainder)
But how do we find n mod 7? Especially when n is something huge like 1000000? That is where we need to make some useful observations about the modulus operator.
x mod a = y mod a, implies, x - y is divisible by a (figure shows why)

What can we say about xⁿ mod a and yⁿ mod a?
If x mod a = y mod a, then xⁿ mod a = yⁿ mod a.
In other words, if x - y is divisible by a then xⁿ-yⁿ is divisible by a too.
xⁿ mod a = yⁿ mod a is true only if xⁿ - yⁿ is divisible by a. Is that the case?
YES! And that is because xⁿ-yⁿ = (x-y)(some factor)
examples:
x²-y² = (x-y)(x+y)
x³-y³ = (x-y)(x² + xy + y²)
and so on.
Since x mod a = y mod a, ie x - y is divisible by a, (x-y)(some factor) is also divisible by a (since the “some factor” has no fractions and is therefore an integer).
OK, so what day is a million days from today?
million days from today = today + 100000 mod 7
how can we solve 1000000 mod 7?
Observe that 10 mod 7 = 3 (or in other words 3 mod 7)
10 mod 7 = 3 mod 7
Sqaring both sides confidently because we proved that exponentiation on both sides retains the equality,
100 mod 7 = 9 mod 7
Reducing right hand expression, by observing 9 mod 7 = 2 = 2 mod 7
100 mod 7 = 2 mod 7
Cubing both sides,
1000000 mod 7 = 8 mod 7
Observing that 8 mod 7 = 1
1000000 mod 7 = 1
a million days from now = today + 1 = Saturday + 1 = Sunday. (Something tells me that no one would worry about weekdays and weekends then)

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Anagram, Anugram, Anigram

February 14th, 2004 admin

I chanced to realize that a signature I had used for my emails at work, “eleven plus two = twelve plus one” is infact what is called an anugram. An anugram is an anagram that also has truth associated with it. “west = stew” is therefore an anagram but not an anugram. Anigram is an animated anagram, that one might use on a webpage to show how anagrams work. The definition of an anigram is therefore restricted, shallow and quite unnecessary. A pangram is a sentence that has all the letters of the english alphabet. I am guessing that for languages like Indian languages, it would be harder to make such a restriction since it is rare to not find a vowel following a consonant. Most often he basic ‘a’ sound as in ‘bus’, is found following every consonant sound. Read up more about anagrams and create your own anagrams at the wordsmith website .

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