Multiple of 72

The seven-digit integer 2718AB6 is a multiple of 72 (where A,B are digits). Determine all possible ordered pairs (A,B).
Source: NCTM Mathematics Teacher, December 2005

SOLUTION
Since 3\times 2=6 and 8\times 2=16, we multiply 72 by 03,13,23, etc. and by 08,18,28, etc. to find the possible AB digits that would match 8AB6
72\times 03=216
72\times 13=936
72\times 23=1656
72\times 33=2376
72\times 43=3096
72\times 53=3816
72\times 63=4536
72\times 73=5256
72\times 83=5976
72\times 93=6696
Only 216 and 936 are close match of 8AB6.
Verification
2718216=72\times 37753
2718936=72\times 37763
We do the same multiplication of 72 by 08,18,28, etc.
72\times 08=576
72\times 18=1296
72\times 28=2016
72\times 38=2736
72\times 48=3456
72\times 58=4176
72\times 68=4896
72\times 78=5616
72\times 88=6336
72\times 98=7056
Only 576 is a close match
Verification
2178576=72\times 30258
The possible ordered pairs (A,B) are (2,1),(9,3),(5,7).

Answer: (2,1),(5,7),(9,3)

Alternative solution
Since 72=8\times 9, 2178AB6 must be divisible by 8 and 9. To be divisible by 8 the last three digits AB6 must be divisible by 8. To be divisible by 9 the sum of the digits must be divisible by 9.
2+1+7+8+A+B+6=24+A+B
27 and 36 are divisible by 9
27=24+A+B
A+B=3
36=24+A+B
A+B=12
Case 1: A+B=3
03, 036 not divisible by 8
12, 126 No
21. 216 Yes
30, 306 No
Case 2: A+B=12
39, 396 not divisible by 8
48, 486 No
57, 576 Yes
66, 666 No
75, 756 No
84, 846 No
93, 936 Yes

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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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