A big number that shrinks down to about three

I remember this from a book.

The puzzle is:

The sum of the factors of 360 is 1170. What is the sum of the reciprocals of the factors of 360?

At first glance, it might seem impossible to solve, but when you look at it deeper, it’s easy.

Let’s do the classic:

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/8 + 1/9 + 1/10 + 1/12 + 1/15 + 1/18 + 1/20 + 1/24 + 1/30 + 1/36 + 1/40 + 1/45 + 1/60 + 1/72 + 1/90 + 1/120 + 1/180 + 1/360 = Exactly 3.25

BUT

Since the factors all share a common multiple of 360, We can do this:

360/360 + 180/360 + … + 2/360 + 1/360, where the numerators are all the factors and the denominator is 360 all the way. But the sum of the numerators is 1170, so the answer should be 1170/360, and it is, which also simplifies to 3.25

Challenge: what is the sum of the reciprocals of the squares of the factors of 360, given that the sum of the square of every factor of 360 is 201110? Yes, I had to calculate it by hand. 😦

Same thing, but just all the reciprocals are squared. 129600/129600 + 32400/129600 + … + 4/129600 + 1/129600 = 201110/129600 ≈ 1.55177469136

Also, I don’t know if this is for everyone right now, but Google has a weird thing where if you press / it jumps to your search bar, so I thought the Google calculator was broken.

One thought on “A big number that shrinks down to about three

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