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Tweedle Dum and Tweedle Dee

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

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.

Unit Fractions

Age 11 to 14 Challenge Level:

Consider the equation ${1\over a} +{1\over b}+{1\over c} = 1$ where $a$, $b$ and $c$ are natural numbers and $0 < a < b < c$. Prove that $a< 3$ and also that $b< 4$ and hence that there is only one set of values which satisfy this equation.

Find the six sets of values which satisfy the equation ${1\over a} +{1\over b}+{1\over c} +{1\over d}= 1$ where $a$, $b$, $c$ and $d$ are natural numbers and $0 < a < b < c < d$.