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#20 Factor-finding mission

Set by Mary Ellis

154 has three prime factors (2, 7 and 11), but it can be divided by other numbers, including 1, 14, 22, 77 and 154. These are factors of 154, but not prime factors.

What is the smallest number that has four distinct prime factors? How many factors does this number have?

What is the next number after this that has the same total number of factors? (Hint: it isn’t a number with four distinct prime factors!)

What is the smallest number that has that same total number of factors?

Solution next week

#19 Angular arrangements

Solution

Each angle in the pentagon’s centre is 360°/5 = 72°. Since a triangle’s angles add to 180°, the two outer angles add up to 180° – 72° = 108°, so the regular pentagon’s interior angles are all 108°. For a polygon with n sides, the centre angles would be 360°/n, so the interior angles would be 180° – (360°/n). The interior angles of sets of regular polygons that meet at a point must add to 360°. We can depict these sets using arrays like {3,3,3,4,4}, which represents three equilateral triangles and two squares. The possibilities are {3,3,3,3,3,3}, {3,3,3,3,6}, {3,3,3,4,4}, {3,3,6,6}, {3,3,4,12}, {3,4,4,6}, {3,12,12}, {3,10,15}, {3,9,18}, {3,8,24}, {3,7,42}, {4,4,4,4}, {4,8,8}, {4,6,12}, {4,5,20}, {5,5,10} and {6,6,6}. The maximum number of sides is 42. A 42-gon with a triangle and heptagon makes 360°. Past this, summing to 360° is impossible.