To determine the smallest number in the form $x123yzt$ that is divisible by $72$, we need to understand the divisibility conditions:

1. Divisibility by 72 means the number must be divisible by both 8 and 9.
   • Divisibility by 8: The last three digits of the number must be divisible by $8$.
   • Divisibility by 9: The sum of all the digits must be divisible by $9$.

Step-by-Step Solution:

1. Given format: $x123yzt$
   Here, $x, y, z, t$ are digits (0–9).

2. Condition 1: Divisibility by 8

   • Focus on the last three digits $yzt$. To satisfy divisibility by 8, $yzt$ must be divisible by 8.

3. Condition 2: Divisibility by 9

   • The sum of the digits $x + 1 + 2 + 3 + y + z + t$ must be divisible by $9$.

Steps to Find the Smallest Number:

1. Assume $x = 0$ to minimize the number.
   Then the number becomes $0123yzt$, which is effectively $123yzt$.
2. Test small values for $yzt$ such that:
   • $yzt$ is divisible by $8$, and
   • The sum $1 + 2 + 3 + y + z + t$ is divisible by $9$.

Testing:

• Start with small values for $yzt$:
  • $yzt = 024$:
    • $024 \div 8 = 3$ (divisible by 8).
    • Sum of digits $1 + 2 + 3 + 0 + 2 + 4 = 12$ (not divisible by 9).
  • $yzt = 072$:
    • $072 \div 8 = 9$ (divisible by 8).
    • Sum of digits $1 + 2 + 3 + 0 + 7 + 2 = 15$ (not divisible by 9).
  • $yzt = 144$:
    • $144 \div 8 = 18$ (divisible by 8).
    • Sum of digits $1 + 2 + 3 + 1 + 4 + 4 = 15$ (not divisible by 9).
  • $yzt = 192$:
    • $192 \div 8 = 24$ (divisible by 8).
    • Sum of digits $1 + 2 + 3 + 1 + 9 + 2 = 18$ (divisible by 9).

Final Answer:

The smallest number in the form $x123yzt$ that is divisible by $72$ is:
$\mathbf{123192}$

Let me know if you want further clarifications or additional examples!

Related Questions:

1. How do you determine if a number is divisible by 8?
2. How can you check for divisibility by 9?
3. What is the general method to solve divisibility problems like this?
4. Can this method be applied to larger numbers?
5. How does the divisibility rule for 72 combine other rules?

Tip: Always start with conditions that restrict the form of the number, like divisibility rules, to simplify your search.