Final answer:
The tens digit of a perfect square with a units digit of 6 can be 1, 5, 7, or 9. This is deduced by squaring numbers ending in 4 or 6 and observing patterns in tens digits of these squares.
Step-by-step explanation:
The possible values of the tens digit for a number whose units digit of a perfect square is 6 can be deduced by examining the pattern of the units digits of perfect squares. If a perfect square ends in 6, we know that it must have resulted from squaring a number that ends in either 4 or 6, as only 4² (which is 16) and 6² (which is 36) have a units digit of 6. To find the tens digit, we need to look at more examples of squares that end in 6:
- 14² = 196
- 24² = 576
- 34² = 1156
- 44² = 1936
- 54² = 2916
- 64² = 4096
- 74² = 5476
- 84² = 7056
- 94² = 8836
From these examples, we see that the possible tens digits are 1, 3, 5, 7, 9, and 0 (considering only the last two digits of each square). However, 2, 4, 6, and 8 do not appear as the tens digit in these examples of perfect squares ending in 6. Therefore, the possible values for the tens digit are 1, 5, 7, and 9, which repeat in a pattern as the ending digits of the base numbers increase. The tens digit depends on which of the base numbers (ending in 4 or 6) is being squared.