Integer Q

Find the greatest integer q such that 475<q<875 and the last three digits of q^3 (ones, tens, and hundreds) are the same as integer q itself.
Source: mathcontest.olemiss.edu 4/29/2013

SOLUTION
Let abc be the three digits of q where a is the hundreds digit, b the tens digit, and c the ones digit.
We want c^2 to end in 1 so that c^2\times c=c.
1^2=1
9^2=81
The ones digit c is either 1 or 9.
(abc)^2 ending in 001 is a necessary condition. For example,
249^2=62001
Multiply 249^2\times 249
\quad\;62001
\times\;249
\text{-------------}
00558009
0248004
124002
\text{-------------}
15438249

249^2=62001\quad\;\;249^3=15438249
251^2=63001\quad\;\;251^3=15813251
499^2=249001\quad 499^3=124251499
501^2=251001\quad 501^3=125751501
749^2=561001\quad 749^3=420189749
751^2=564001\quad 751^3=423564751
999^2=998001\quad 999^3=997002999

751 is the largest integer such that 475<751<875 and the last three digits of 751^3 match the digits of 751

Answer: 751

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About mvtrinh

Retired high school math teacher.
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