Divisibility By 3 & 6: Find The Missing Digits!

Hey guys! Let's dive into the fascinating world of divisibility and tackle a fun challenge: figuring out the missing digits in numbers to make them divisible by 3 and 6. This is a crucial concept in mathematics, and mastering it can help you simplify calculations and understand number patterns better. So, grab your pencils, and let's get started!

Divisibility by 3: Cracking the Code

So, you're wondering how to determine if a number is divisible by 3, right? Well, here's the secret: a number is divisible by 3 if the sum of its digits is divisible by 3. Let's break that down. To make these numbers divisible by 3, we will apply this rule and find the missing digit. We'll go through several examples together to make sure you grasp this concept. It’s easier than you think, I promise!

Let's start with the first set of examples where we need to find the missing digit to make the number divisible by 3:

Examples for Divisibility by 3

a) 3.19: To figure out the missing digit, let's add the digits we have: 3 + 1 + 9 = 13. Now, what number do we need to add to 13 to get a multiple of 3? The next multiple of 3 after 13 is 15, so we need to add 2 (15 - 13 = 2). Therefore, the missing digit is 2, and the number becomes 3219. Isn't that cool?

b) 3 4 56: Let's do the same thing here. Add the digits: 3 + 4 + 5 + 6 = 18. Guess what? 18 is already divisible by 3! That means we can put a 0 in the blank (30456) or any multiple of 3 (like 3, 6, or 9) and the number will still be divisible by 3. Let's go with 0 for simplicity. The number is 30456.

c) 5 21: Add the digits: 5 + 2 + 1 = 8. The next multiple of 3 after 8 is 9, so we need to add 1. The missing digit is 1, making the number 5121.

d) 42: This one’s a bit trickier because the blank is in a different spot, but the principle is the same. Add the digits: 4 + 2 = 6. Again, 6 is already divisible by 3, so we can use 0 as the missing digit to get 402.

e) 54 2: Adding the known digits: 5 + 4 + 2 = 11. The next multiple of 3 is 12, so we need to add 1. The number becomes 5412.

f) 5 020: Here we have 5 + 0 + 2 + 0 = 7. We need to add 2 to get to 9, which is divisible by 3. The number is 52020.

g) 52 2 : 9: Okay, this one looks a bit different with that 9 at the end, but don't worry! We add the digits: 5 + 2 + 2 + 9 = 18. Since 18 is already divisible by 3, we can use 0. The number is 52029.

h) 4 79: Add those digits: 4 + 7 + 9 = 20. The next multiple of 3 after 20 is 21, so we add 1. The number is 4179.

i) 34: Adding the digits: 3 + 4 = 7. We need to add 2 to get to 9. The number is 324.

j) 61 23: Summing the digits: 6 + 1 + 2 + 3 = 12. Guess what? 12 is divisible by 3! So, we use 0. The number is 61023.

k) 1 : 8311 12: This looks long, but we've got this! 1 + 8 + 3 + 1 + 1 + 1 + 2 = 17. The next multiple of 3 is 18, so we add 1. The number is 11831112.

l) 14: Adding the digits: 1 + 4 = 5. We need to add 1 to reach 6. So, the number is 114.

m) 47 17: Let's add: 4 + 7 + 1 + 7 = 19. We need to add 2 to get to 21. The number is 47217.

n) 21336: Sum the digits: 2 + 1 + 3 + 3 + 6 = 15. It’s already divisible by 3, so we use 0. The number is 210336.

o) 38: Adding the digits: 3 + 8 = 11. We need to add 1 to get to 12. The number is 318.

p) 2 32: Adding: 2 + 3 + 2 = 7. We add 2 to get to 9. The number is 2232.

q) 517 498: Sum them up: 5 + 1 + 7 + 4 + 9 + 8 = 34. We need to add 2 to get to 36. The number is 5172498.

r) 7 708: Adding: 7 + 7 + 0 + 8 = 22. We need to add 1 to get to 24. The number is 71708.

See? It's all about finding that missing piece to make the sum of the digits a multiple of 3.

Divisibility by 6: The 2-and-3 Rule

Now, let's move on to divisibility by 6. This one’s super cool because it combines what we already know about divisibility by 2 and 3. Here’s the rule: a number is divisible by 6 if it’s divisible by both 2 and 3. This means the number must be even (divisible by 2) and the sum of its digits must be divisible by 3 (divisible by 3).

Let's tackle some examples to find the missing digits for divisibility by 6. Remember, we need to ensure the number is even and the sum of the digits is divisible by 3. Let’s do this!

Examples for Divisibility by 6

a) 2 6: First, we need the number to be even, so the missing digit has to be even (0, 2, 4, 6, or 8). Let's add the known digits: 2 + 6 = 8. Now, we need to find an even number that, when added to 8, gives us a multiple of 3. If we add 4, we get 12, which is divisible by 3. So, the missing digit is 4, making the number 246.

b) 9 : The missing digit has to be even. Let’s think… 9 plus what even number gives us a multiple of 3? If we add 0, we get 9, which is divisible by 3. So, the missing digit is 0, and the number is 90.

c) 7 : Again, we need an even number. 7 plus what even number is a multiple of 3? If we add 2, we get 9. So, the missing digit is 2, and the number is 72.

d) 42 : This one is interesting! 4 + 2 = 6, which is divisible by 3. The missing digit needs to be even, and adding 0 works perfectly since 420 is even and the sum of digits is still 6. The number is 420.

e) 27 : The missing digit must be even. 2 + 7 = 9, which is divisible by 3. Using 0 as the missing digit gives us 270, which is even. So, the number is 270.

f) 6 : An even number needed here. 6 plus what even number is divisible by 3? We can use 0, making the number 60.

g) 41 60: Adding the digits we have: 4 + 1 + 6 + 0 = 11. We need an even digit that, when added to 11, makes a multiple of 3. If we add 1, we don’t get an even number. If we add 4 to the sum, 11 + 4 = 15, which is divisible by 3. However, 4 is not an even digit and 41460 is not divisible by 6. So let's try 11 + x = multiple of 3, where x is a single digit. We can reach 12 by adding 1 to 11. But we can also reach 15 by adding 4 to 11. But we also need the number to be divisible by 2, so the last digit should be even. So x can be 4. So 41460 works as 41460/6 = 6910. The number is 41460.

h) 38 : The missing digit must be even. 3 + 8 = 11. We need an even number to add to 11 to get a multiple of 3. Adding 4 gives us 15, which works! The number is 384.

i) 81 : The missing digit must be even. 8 + 1 = 9, which is divisible by 3. Adding 0 makes it 810, which is even. So, the number is 810.

Awesome! We’ve navigated through divisibility by 6 by making sure our numbers follow both the rules for 2 and 3. You’re becoming divisibility experts!

Wrapping Up: You're Divisibility Wizards!

So there you have it, guys! We've unlocked the secrets of divisibility by 3 and 6. Remember, for 3, it’s all about the sum of the digits. And for 6, think 2 and 3 together – even numbers with digits that add up to a multiple of 3. Keep practicing, and you'll become a pro at spotting these divisibility patterns. You got this!

If you have any questions or want to try more examples, just let me know. Keep up the great work, and happy calculating!