Fives and Eights

Find the greatest positive integer that cannot be expressed as a multiple of 5, as a multiple of 8, or as a sum of a multiple of 5 and a multiple of 8.
Source: mathcontest.olemiss.edu 3/24/2014

SOLUTION
What kind of numbers satisfy the requirements? We look at numbers that are not multiple of 5 or not multiple of 8 and try to express them as a sum of a multiple of 5 and a multiple of 8. If we cannot, the numbers satisfy the requirements.
6=5+1\qquad\mathbf{Yes}
7=5+2\qquad\mathbf{Y}
9=5+4\qquad\mathbf{Y}
=8+1
11=5+6\qquad\mathbf{Y}
=10+1
=8+3
12=5+7\qquad\mathbf{Y}
=10+2
=8+4
13=5+8\qquad\mathbf{No}
14=5+9\qquad\mathbf{Y}
=10+4
=8+6
17=5+12\qquad\mathbf{Y}
=10+7
=15+2
=8+9
=16+1
18=5+13\qquad\mathbf{N}
=10+8
19=5+14\qquad\mathbf{Y}
=10+9
=15+4
=8+11
=16+3
21=5+16\qquad\mathbf{N}
22=5+17\qquad\mathbf{Y}
=10+12
=15+7
=20+2
=8+14
=16+6
23=5+18\qquad\mathbf{N}
=10+13
=15+8
26=5+21\qquad\mathbf{N}
=10+16
27=5+22\qquad\mathbf{Y}
=10+17
=15+12
=20+7
=25+2
=8+19
=16+11
=24+3
From 28 on all numbers not ending in 0 or 5 can be expressed as a sum of a multiple of 5 and a multiple of 8 by the following rules.
If the number ends in 1, subtract 16 and get a sum = xxx5+16
If the number ends in 2, subtract 32 and get a sum = xxx0+32
If the number ends in 3, subtract 8 and get a sum = xxx5+8
If the number ends in 4, subtract 24 and get a sum = xxx0+24
If the number ends in 6, subtract 16 and get a sum = xxx0+16
If the number ends in 7, subtract 32 and get a sum = xxx5+32
If the number ends in 8, subtract 8 and get a sum = xxx0+8
If the number ends in 9, subtract 24 and get a sum = xxx5+24
27 is the greatest positive integer that cannot be expressed as a multiple of 5, as a multiple of 8, or as a sum of a multiple of 5 and a multiple of 8.

Answer: 27

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About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , . Bookmark the permalink.

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