Accidentally Right?

Number Theory Level 3

$\large \frac{163}{326} = \frac{1 \cancel{6} \cancel{3}}{ \cancel{3} 2 \cancel{6}} = \frac{1}{2}$

Strangely, this actually works. $163 \div 326$ does in fact equal $\frac{1}{2}$ . How many other fractions have this same property? (Namely, if you cancel out like numbers in the numerator and denominator, you are left with a fraction that is equal to the original fraction?)

Greater than 20, but less than 30 Greater than 0, but less than 10 Greater than 10, but less than 20 Greater than 30

1 solution

Joshua Lowrance
Jan 15, 2019

In fact, there exists an infinite number of such fractions:

$\large \frac{16}{64} = \frac{166}{664} = \frac{1666}{6664} = \frac{16666}{66664} = \cdots = \frac{1}{4}$

All of the $6$ 's cancel out, leaving $\frac{1}{4}$ . $166\cdots 66 \div 66\cdots 664$ always equals $\frac{1}{4}$ . I challenge you to figure out why this is so (it's actually not that difficult, just multiply $166\cdots 66$ by $4$ ).

$\frac {10}{20} = \frac {100}{200} = \frac {1000}{2000} = \cdots = \frac 12$

is another example of such an infinite sequence.

And in fact, there is an infinite number of such infinite sequences because instead of $2000\cdots00$ you could write any number that has the right number of 0's and doesn't contain any 1's.

Henry U - 2 years, 4 months ago

Yep! I feel like it is easy to forget these cases because it doesn't seem natural to cancel out $0$ 's like this. Great job catching this!

Joshua Lowrance - 2 years, 4 months ago

An infinite number of infinite sequences is just an infinite sequence. More formally the cartesian product of two sets with cardinality aleph 0(examples include the set of rational numbers or the set of integers) is a set that also has cardinality aleph 0. So having an infinite number of sets with an infinite number of terms is just an infinite set with the same size.

Razzi Masroor - 1 year, 4 months ago

Accidentally Right?

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