Question

question 3 of 10
if number a is a 2-digit number and its digits are transposed to form number
b, then the difference between the larger of the two numbers and the smaller
of the two numbers must be divisible by ______.

a. an even number
b. 9
c. 7
d. 8

Explanation:

Step1: Represent the two - digit number

Let the tens digit of number A be \(a\) and the units digit be \(b\), where \(a\) and \(b\) are non - negative integers and \(1\leqslant a\leqslant9\), \(0\leqslant b\leqslant9\). Then number A can be written in the form of \(10a + b\). When we transpose the digits to form number B, number B is \(10b + a\).

Step2: Calculate the difference between the two numbers

Without loss of generality, assume \(a>b\) (if \(b > a\), we can just swap the two numbers in the subtraction). Then the difference between the larger number and the smaller number is \((10a + b)-(10b + a)\).
Simplify the expression:

$$
LATEXBLOCK0
$$

Since the difference is of the form \(9(a - b)\) (where \(a - b\) is an integer), the difference between the larger number and the smaller number must be divisible by 9.

Answer:

B. 9