Rearrange the Digits 2, 3, and 5 to Create a Number Divisible by 10

Q: Why can't I just put 5 at the end since 5 divides into 10?

Great thinking, but divisibility by $10$ requires the number to end in 0, not $5$. Numbers ending in $5$ are divisible by $5$, but not by $10$.

Q: What if the question asked about divisibility by 5 instead?

Then we could solve it! Numbers divisible by $5$ must end in $0$ or $5$. Since we have the digit $5$, arrangements like 235 and 325 would work.

Q: Are there other digits that would make this impossible?

Yes! If you need a number divisible by $10$ but don't have the digit $0$, it's always impossible. The divisibility rule for $10$ is non-negotiable.

Q: How do I quickly recognize when a rearrangement problem is impossible?

Check the divisibility rule first! For divisibility by $10$, you need $0$. For even numbers, you need $0, 2, 4, 6,$ or $8$ at the end. No required digit = impossible!

Q: Could I add a 0 to make this work?

Only if the problem allows it! This question specifically says to use only the digits $2, 3,$ and $5$. Adding extra digits would change the problem entirely.

Divisibility Rules with Missing Digits

Rearrange the following digits to create a number divisible by 10:

2, 3 , and 5.

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Step-by-step video solution

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00:00 Compose a number that is divisible by 10

00:04 A number divisible by 10 is a number with units digit 0

00:07 Using this method, we will go through all numbers and eliminate accordingly

00:16 We can see there is no suitable number, and this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Rearrange the following digits to create a number divisible by 10:

2, 3 , and 5.

Step-by-step solution

To solve this problem, we need to understand the divisibility rule for $10$ . A number is divisible by $10$ if and only if it ends in $0$ .

Given the digits $2$ , $3$ , and $5$ , we need to form a number ending with $0$ . However, none of these digits is $0$ . Therefore, using only the digits $2$ , $3$ , and $5$ , it is impossible to create a number that ends in $0$ .

This means it's impossible to rearrange these digits to form a number divisible by $10$ .

Therefore, the correct answer is It is impossible.

Final Answer

It is impossible.

Key Points to Remember

Essential concepts to master this topic

Rule: Numbers divisible by 10 must end in 0
Technique: Check if digit 0 exists in given set: 2, 3, 5
Check: No 0 available means no valid arrangement possible ✓

Common Mistakes

Avoid these frequent errors

Testing all possible arrangements without checking divisibility rule first
Don't try arranging 235, 253, 325, 352, 523, 532 and testing each = wasted time! This ignores the fundamental rule. Always check if the required ending digit (0 for divisibility by 10) is available first.

Practice Quiz

Test your knowledge with interactive questions

Is the number 10 divisible by 4?

FAQ

Everything you need to know about this question

Why can't I just put 5 at the end since 5 divides into 10?

Great thinking, but divisibility by $10$ requires the number to end in 0, not $5$ . Numbers ending in $5$ are divisible by $5$ , but not by $10$ .

What if the question asked about divisibility by 5 instead?

Then we could solve it! Numbers divisible by $5$ must end in $0$ or $5$ . Since we have the digit $5$ , arrangements like 235 and 325 would work.

Are there other digits that would make this impossible?

Yes! If you need a number divisible by $10$ but don't have the digit $0$ , it's always impossible. The divisibility rule for $10$ is non-negotiable.

How do I quickly recognize when a rearrangement problem is impossible?

Check the divisibility rule first! For divisibility by $10$ , you need $0$ . For even numbers, you need $0, 2, 4, 6,$ or $8$ at the end. No required digit = impossible!

Could I add a 0 to make this work?

Only if the problem allows it! This question specifically says to use only the digits $2, 3,$ and $5$ . Adding extra digits would change the problem entirely.

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