2,3,4,5,6,7 AND 8 EXACTLY ONCE IN THEM. DETERMINE THIS INTEGER. 【SO...
1,2,3,4,5,6,7 and 8 exactly once in them. Determine this integer.
【Solution】
Since 20
2
= 400 and 20
3
= 8000 together use only 7 digits, our number is greater
than 20. Since 32
2
= 1024 and 32
3
= 32768 together use 9 digits, our number is
less than 32. The units digit of our number cannot be 0, 1, 5 or 6 as otherwise both the
square and the cube end in the same digit. It cannot be 3 or 7 as otherwise the square
ends in 9. Hence our number is 22, 24 or 28.
Since 22
2
= 484 , it is not 22. Since 28
3
= 21952 , it is not 28. Hence it must be 24,
and indeed 24
2
= 576 and 24
3