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Here's An interesting experiment - lay out 13 cards of a single suit face
down in order Ace, 2, 3, ...Queen, King. Starting with card 1, turn over every
card. Then starting with card 2, turn over every 2nd card, then starting with
card 3 turn over every 3rd card, etc., until you turn over just the 13th card on
the 13th pass.
After 6 passes
Which cards will be face up after 13 passes? Can you guess which cards would
be face up if we had cards numbered 1 to 50? Can you explain why? (Spoiler
at bottom of page.)
Note that each card gets flipped a number of times related to the number of
divisors that it has. That should be enough of a hint to get you
moving toward answering the above questions.
By the way, here is a general method for determining the number of unique
divisors of a number without listing them all: How many divisors are
there for 72?
|Write the prime factorization of the number, 72=23 X 32|
|Add 1 to the exponents of each of the factors and multiply the
results. The answer is d(N), the number of unique divisors of the
original number N. For our example, d(72)=
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|Since all cards started face down, any that land face up
will have been flipped an odd number of times. And, since each flip
represents a divisor, the face up cards have an odd number of divisors. .
These will always be perfect squares because squares are the only
numbers with an odd number of divisors. This is clear if we notice
that divisors of a number, N, always occur in pairs (the
divisor, D, and the quotient, Q satisfy N = D x
Q ). Thus each new divisor will increase the total number of
unique divisors by two, unless the divisor and the quotient are the same
. in which case the number is a perfect square and number
of divisors increases by one to an odd number.|