Multiples of 11

Every once in a while you’ll need to test a number for divisibility by 11.  Multiples of 11 can be quickly identified if the alternating sum of their digits is a multiple of 11.  This works because the consecutive powers of 10 are either one more or one less than a multiple of 11.  For example, 10 is one less than a multiple of 11, and 100 is one more than a multiple of 11, and 1000 is one less than a multiple of 11.

If you pull out these multiples of 11 from a number, then you are left with an alternating sum.  Since the sum of 2 multiples of 11 is a multiple of 11, then the alternating sum is also a multiple of 11.

This is clearer in my notes, where a, b, c, and d are the digits of a 3 and 4 digit number:IMG_20171121_141127.jpg

 

Trapezoid existence

Not all 3 line segments will form a triangle.  If you don’t believe me, try forming a triangle using line segments of lengths 2, 2 and 20.  The two sides of length 2 aren’t long enough to span the 3rd side.  A quick test for this is to see that each side is smaller than the combined lengths of the other 2 sides.  And that each side is greater than the difference between the other 2 sides.

This week I learned that there is also a fun test to determine whether 4 given side lengths, when assigned to be either the parallel or non-parallel sides can actually form a trapezoid.  Not all 4 side lengths in a particular orientation can form a trapezoid.  The formula seems weird, but it’s actually an extension of the triangle inequality.  Here a and b are the lengths of the parallel sides and c and d are the non-parallel sides.

|d-c|<|b-a|<d+c

To form the triangle, draw a line from a vertex parallel to a non-parallel side to the opposite parallel side (see below).  The triangle has side lengths b-a, c, and d.

An old AMC gives 4 side lengths of possible trapezoids, and the student must test each orientation to determine whether a trapezoid can be formed.

IMG_20171128_141730.jpg

Operation

In the old game of Operation, the surgeon/player could earn $100 by removing Butterflies in the Stomach, an actual butterfly from the patient’s torso.   But there’s no need to feel butterflies when you see a strange operation defined for the symbol:  ¶.

I like to tell my students they can define any symbol to represent any operation they can dream up.

a \odot b = a-b+17

a \star b = -37

This AMC problem adds a bit of complexity as it’s an operation that is defined for 3 inputs.

IMG_20171121_143233.jpg