Math Problem Statement

what is the smallest number in the form x123yzt that is divisible by 72?

Solution

To solve the problem, we need to find the smallest number of the form x123yztx123yzt that is divisible by 72.

Divisibility by 72

72 can be factored into prime factors as:

72=8×972 = 8 \times 9

This means that for a number to be divisible by 72, it must be divisible by both 8 and 9.

Divisibility Rule for 8

A number is divisible by 8 if its last three digits are divisible by 8. In this case, the last three digits of the number are yztyzt, so we need:

yzt divisible by 8yzt \text{ divisible by } 8

Divisibility Rule for 9

A number is divisible by 9 if the sum of its digits is divisible by 9. The digits of the number are x,1,2,3,y,z,tx, 1, 2, 3, y, z, t, so we need:

x+1+2+3+y+z+t divisible by 9x + 1 + 2 + 3 + y + z + t \text{ divisible by } 9

This simplifies to:

x+6+y+z+t divisible by 9x + 6 + y + z + t \text{ divisible by } 9

Step-by-Step Solution

  1. Start with the divisibility by 8: The number yztyzt must be divisible by 8. We will check values of yztyzt from 000 to 999 to find the smallest one that satisfies divisibility by 8.

  2. Ensure divisibility by 9: After determining a valid yztyzt, we will check the condition for divisibility by 9 by adjusting xx accordingly.

Finding the Smallest Number

Let's go through these steps:

  1. Divisibility by 8: We need to find the smallest yztyzt such that yztyzt is divisible by 8. Starting from 000, the smallest value for yztyzt divisible by 8 is 000 (since 000 is divisible by 8).

  2. Divisibility by 9: Now, substitute yzt=000yzt = 000 into the sum of the digits condition. The sum of the digits becomes:

    x+6+0+0+0=x+6x + 6 + 0 + 0 + 0 = x + 6

    For this sum to be divisible by 9, x+60(mod9)x + 6 \equiv 0 \pmod{9}, which means x3(mod9)x \equiv 3 \pmod{9}. The smallest such value of xx is x=3x = 3.

Thus, the smallest number in the form x123yztx123yzt that is divisible by 72 is 3123000.

Summary:

The smallest number in the form x123yztx123yzt that is divisible by 72 is 3123000.

Would you like a more detailed explanation of the divisibility rules, or do you have any further questions on this? Here are some related questions:

  1. How can you test whether a larger number is divisible by 72?
  2. What are some common divisibility rules for other numbers?
  3. How does the concept of modulo arithmetic help in divisibility problems?
  4. How can you solve similar problems with different forms of numbers?
  5. What are the properties of numbers divisible by both 8 and 9?

Tip: For divisibility by 8, only the last three digits matter, and for divisibility by 9, the sum of the digits must be divisible by 9.

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Math Problem Analysis

Mathematical Concepts

Divisibility Rules
Prime Factorization
Modulo Arithmetic

Formulas

Divisibility rule for 8: last three digits divisible by 8
Divisibility rule for 9: sum of digits divisible by 9

Theorems

Prime Factorization Theorem
Divisibility by 72 (8 x 9)

Suitable Grade Level

Grades 6-8