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#58 Fulfilling fractions

Set by Mary Ellis

Arrange the numbers 1, 2, 3, 4, 5 and 6 into three fractions. Choose two of the fractions and multiply them together, then add the third fraction. What is the largest number you can make?

Using the fractions 0/7, 3/9, 2/5, 8/4 and 6/1, and the operations +, –, × and ÷ once each (and brackets as needed), can you make a value of 7?

It is possible to make a number as big as 1512 using the digits 0-9 and those four operations once each. Fit the digits 0-9 into this equation, replacing the dots, to make it true:

Solution next week

#57 Prime jumps

Solution

You must add 100 to 21 four times, giving 421, to reach a prime number. A starting number will never reach a prime if it is a multiple of 2 or 5, since it will have a common factor with 100. That leaves 40 numbers – those ending in 1, 3, 7 or 9.

Of these, 23 are prime. 27, 51, 69, 77, 81, 93, 97 all give non-primes when adding 400, and of the remainder only one has the required property: 87 = 3 × 29, 187 = 11 × 17, 287 = 7 × 41 and 387 = 3 × 129, but 487 is prime.

If we relax the requirement to start with a non-prime, then 19, 43 and 61 are added to our list, needing to get to 419, 443 and 461 to find another prime.