Which of the following numbers is divisible by 22?

The correct option is 893002.

To determine which number is divisible by 22, we must check the divisibility rules for the prime factors of 22, which are 2 and 11 (since $22=2\times 11$). A number is divisible by 22 if and only if it is divisible by both 2 and 11.

1. Rule for Divisibility by 2:

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

Let's check the given options:

• 645372: Ends in 2 (Even) - Divisible by 2

• 893002: Ends in 2 (Even) - Divisible by 2

• 654320: Ends in 0 (Even) - Divisible by 2

• 602351: Ends in 1 (Odd) - Not divisible by 2 (Eliminated)

2. Rule for Divisibility by 11:

A number is divisible by 11 if the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) is either 0 or a multiple of 11.

Let's test the remaining even numbers:

Testing 893002:

Digits at odd places (1^st, 3^rd, 5^th from right): 2, 0, 9

$\text{Sum of digits at odd places}=2+0+9=11$

Digits at even places (2^nd, 4^th, 6^th from right): 0, 3, 8

$\text{Sum of digits at even places}=0+3+8=11$

$\text{Difference}=11-11=0$

Since the difference is 0, 893002 is divisible by 11 as well as 2. Therefore, it is divisible by 22.