AB is a two-digit number where *A* is the digit in the tens place and *B* is the digit in the units place. The product of *A* and *B* is zero. The unit digits of $A^3$, $A^2$, and $A$ are the same, where *A* is the units digit of the given number. Which of the following options contains values nearest to the possible values of the two-digit number AB?

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Accepted Answer

**Solution:**

1. Let the two-digit number be AB, where *A* is the tens digit and *B* is the units digit.

2. Given that the product of *A* and *B* is zero:
   - This means either *A = 0* or *B = 0*.
   - Since AB is a two-digit number, *A* cannot be zero (otherwise it would be a one-digit number).
   - Therefore, *B = 0*.

3. So the number is of the form *A0* where *A* is from 1 to 9.

4. Next, the unit digits of $A^3$, $A^2$, and $A$ are the same, where *A* is the units digit of the given number.
   - Since the units digit of the number is *B = 0*, but the problem states "*A* is the units digit of the given number" for this condition, it implies the problem means the digit *A* (tens digit) is considered for this property.

5. We need to find *A* such that the units digit of $A^3$, $A^2$, and $A$ are the same.

6. Check digits 1 to 9:
   - For 1: units digit of $1^3=1$, $1^2=1$, and $1=1$ → all same
   - For 2: units digit of $2^3=8$, $2^2=4$, and $2=2$ → not same
   - For 3: units digit of $3^3=7$, $3^2=9$, and $3=3$ → not same
   - For 4: units digit of $4^3=4$, $4^2=6$, and $4=4$ → not same
   - For 5: units digit of $5^3=5$, $5^2=5$, and $5=5$ → all same
   - For 6: units digit of $6^3=6$, $6^2=6$, and $6=6$ → all same
   - For 7: units digit of $7^3=3$, $7^2=9$, and $7=7$ → not same
   - For 8: units digit of $8^3=2$, $8^2=4$, and $8=8$ → not same
   - For 9: units digit of $9^3=9$, $9^2=1$, and $9=9$ → not same

7. So possible values of *A* are 1, 5, and 6.

8. Since *B = 0*, the possible two-digit numbers are 10, 50, and 60.

9. Now, check which option contains numbers nearest to these values:
   - Option (a): 9, 49, 64 → 9 is close to 10, 49 close to 50, 64 close to 60
   - Option (b): 9, 36, 100 → 9 close to 10, 36 less close to 50 or 60, 100 is three-digit
   - Option (c): 64, 81, 121 → 64 close to 60, 81 and 121 are not close
   - Option (d): 196, 121, 16 → all far from 10, 50, 60

10. Among these, option (a) seems closest, but 49 and 64 are not exactly near 50 and 60 as 36 and 100 are not near either.

11. However, the question asks for the nearest possible values of the two-digit number AB, and the only valid two-digit numbers are 10, 50, and 60.

12. Since 9, 36, and 100 are given in option (b), and 100 is not two-digit, option (a) with 9, 49, and 64 is the closest set to 10, 50, and 60.

**Answer:** Option (a) 9, 49 & 64

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