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#13 Number Venns

Set by Peter Rowlett

Numbers 1–30 are put in circle A if they are part of group A, in circle B if they belong to group B, in the intersection between if they are part of both A and B, or outside the circles if they don’t belong to either A or B. What are groups A and B for each diagram below (not all numbers have been placed yet)?

Only one of these diagrams can have another number added to the overlapping section using these rules. Which one is it and what is the number?

Solution next week

#12 Factorial factory

Solution

A zero will be present at the end of a number if it is divisible by 2 and 5, as this means it is divisible by 10. If a number has two factors of 5 and 2, it will be divisible by 100 and have two zeros on the end. When making factorials, we multiply numbers together. Each time we multiply by 5 or a number divisible by 5, we introduce a factor of 5. As every second number we multiply by is even, we will always have enough factors of 2 to pair with the 5s – so the number of zeros on the end is determined by the number of factors of 5. For 10!, we multiply by 5 and 10, so have two factors of 5, giving two zeros on the end. 25! has six zeros (multiplying by 25 gives us two factors of 5), and 1066! has 264 zeros (three 5s for 125 and four for 625).