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Tweedle Dum and Tweedle Dee

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Sum Equals Product

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 � 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

Special Sums and Products

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Mathematical Swimmer

Age 11 to 14
Challenge Level

Every day I go to the swimming pool and swim the same number of lengths. I like to count the number of lengths I've done as I go as a fraction of the total number of lengths I'm going to do that day.

If I swam ten lengths a day...

  • After five lengths I would say to myself, "I've managed $\frac{5}{10}$ of my day's swimming, which simplifies to $\frac{1}{2}$."
  • After seven lengths, I would say "I've managed to swim a prime number of lengths." 
  • After eight lengths, I would say "I've done $\frac{8}{10}$ of my day's swimming, which simplifies to $\frac{4}{5}$."
  • After nine lengths, I'd say "I've done 9 lengths, which isn't prime, and $\frac{9}{10}$ does not simplify."

I don't swim ten lengths a day. In fact, the total number of lengths I swim each day is rather special...

Let's call the total number of lengths I swim $n$.

After the first length, for every length I swim, the total so far (let's call it $t$) is either a prime number, or the fraction $\frac{t}{n}$ will simplify (or both).

It is, in fact, the largest number for which this is true.

Can you work out how many lengths I swim each day?