a Choose any four digits. b If it is possible, arrange your digits to make a number that is divisible by † 2 ii 3 iii 4 iv 5 6 c Can you arrange your digits to make a number that is divisible by all five numbers in part a? If not, can you make a number thet is divisible by four of the numbers? d Give your answers to a partner to check.

Let's go through the problem step by step. ### Step 1: Choose Four Digits Let's choose the digits: \(2, 3, 4, 5\). ### Step 2: Check Divisibility Conditions We need to check if we can arrange these digits to form a number that is divisible by the following: #### i. Divisible by 2 A number is divisible by 2 if its last digit is even. In our case, the even digits are \(2\) and \(4\). Therefore, we can end our number with either \(2\) or \(4\). #### ii. Divisible by 3 A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of our digits is: \[ 2 + 3 + 4 + 5 = 14 \] Since \(14\) is not divisible by \(3\), we cannot form a number that is divisible by \(3\) using all four digits. #### iii. Divisible by 4 A number is divisible by 4 if the last two digits form a number that is divisible by 4. Possible pairs of last two digits from our digits are \(24, 32, 52, 54, 42, 34\). We can check: - \(24\) is divisible by \(4\) - \(32\) is divisible by \(4\) - \(52\) is not divisible by \(4\) - \(54\) is not divisible by \(4\) - \(42\) is not divisible by \(4\) - \(34\) is not divisible by \(4\) So, we can use \(24\) or \(32\) as the last two digits. #### iv. Divisible by 5 A number is divisible by 5 if its last digit is \(0\) or \(5\). Since \(5\) is one of our digits, we can end our number with \(5\). #### v. Divisible by 6 A number is divisible by 6 if it is divisible by both \(2\) and \(3\). Since we already established that we cannot form a number divisible by \(3\) with all four digits, we cannot have a number that is divisible by \(6\) using all four digits. ### Step 3: Can We Make a Number Divisible by All Five? Since we cannot make a number that is divisible by \(3\) and \(6\) using all four digits, we cannot make a number that is divisible by all five numbers. ### Step 4: Can We Make a Number Divisible by Four of the Numbers? We can try to make a number that is divisible by \(2, 4, 5\), and \(6\) (but not \(3\)). Let's try the arrangement \(2405\): - **Divisible by 2**: Yes, last digit is \(5\) (not valid). - **Divisible by 4**: Last two digits \(05\) (not valid). - **Divisible by 5**: Yes, last digit is \(5\) (valid). - **Divisible by 6**: Not valid since it is not divisible by \(3\). Now let's try \(4320\): - **Divisible by 2**: Yes, last digit is \(0\). - **Divisible by 3**: Yes, \(4 + 3 + 2 + 0 = 9\) (valid). - **Divisible by 4**: Yes, last two digits \(20\) (valid). - **Divisible by 5**: No, last digit is \(0\) (not valid). - **Divisible by 6**: Yes, since it is divisible by \(2\) and \(3\). ### Conclusion We cannot create a number that is divisible by all five numbers. However, we can create a number that is divisible by \(2, 4, 5\), and \(6\) using the digits \(4, 3, 2, 0\) (e.g., \(4320\)). You can share these findings with a partner for verification!