## ________________

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### Ten From Two

The coach shows the figure below to the team, which demonstrates how to draw a straight line through a pentagram to form seven non-overlapping triangles, as indicated by the dots.

*The team’s challenge: Draw two straight lines through a pentagram to produce ten non-overlapping triangles.*

### Triangular Boundary

The coach then shows the team the diagram below and asks:

*What is the maximum area of a rectangle contained entirely within a triangle with sides of 9, 10, and 17?*

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## Solutions to week 106

The **Simple Multiple** is **10010** and the value for **Small and Simple** is **10 ^{10}, or 10,000,000,000**.

**Simple Multiple answer explained:** 14 = 7 x 2, so any simple integer divisible by 14 must be even, which means it must end with a 0 (rather than a 1). It remains to find a simple integer which is a multiple of 7. Try to divide

1,10,11, 100, etc. by 7 and you will find that 1001 is the first simple integer that is a multiple of 7; so ten times that, namely 10010, is the first multiple of 14. (You can do this a bit faster if you know the special tests for divisibility by 7.)

**Small and Simple answer explained:** Note that any number divisible by 9 will have a “digital sum” (the sum of its digits) divisible by 9. Also, any number divisible by 5 has a final digit of 0 or 5. Since 45 is divisible by both 9 and 5, the digital sum of any simple integer divisible by 45 must end in 0 and have a digital sum divisible by 9. The smallest such simple integer is 1111111110, the next smallest is 10111111110 and so forth until the 10th smallest is 11111111100. The sum of these ten numbers is 99999999990; their average plus 1 is 10^{10}.

## Recent Weeks

**Week 106**: Simple Multiple & Small and Simple, solutions to Math for the Ages & Three Racers

**Week 105**: Math for the Ages & Three Racers, solutions to X Factor & Perfect Pairings

**Week 104**: X Factor & Perfect Pairings, solutions to Two Sums & Double-Faced

**Week 103**: Two Sums & Double-Faced, solutions to Zombie Epidemic & Revenge on the Zombies

**Week 102**: Zombie Epidemic & Revenge on the Zombies, solutions to Heads Up & Balanced Flip

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

**Come back next week** for answers and more puzzles.