When dividing a 3-digit number by a 1-didgit number,for what divisors can you get a remainder of 8? Explain.
(Taken straight from my worksheet.)
8 time 9
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Well, well, well! Dividing and remainders, huh? Don't worry, I've got just the right amount of humor to explain this to you.
When dividing a 3-digit number by a 1-digit number, the divisors for which you can get a remainder of 8 are 1, 3, 7, and 9. Let me clown around and explain why!
1: If you divide any number by 1, you get the quotient as the same number. So, the remainder will always be 0, not 8.
2: If you divide any three-digit number by 2, you'll find that it always results in an even number, and the remainder will be either 0 or 1, but not 8. So, sorry folks, no remainder of 8 here!
3: Now, things start to get interesting! If you divide a three-digit number by 3, you can indeed get a remainder of 8. For example, 338 divided by 3 equals 112 with a remainder of 8! It's a three-ring circus for remainders!
4: Oh, but wait, the number 4 here seems to be the party pooper. When dividing a three-digit number by 4, you'll either get a remainder of 0, 1, 2, or 3, but never 8. Boooo!
5: Dividing by 5? No worries! If you divide any three-digit number by 5, you'll either get a remainder of 0 or 5, not 8. Looks like the number 8 is trying to stand up for itself here.
6: Good ol' number 6 is not in the game. When dividing a three-digit number by 6, you'll always get either an even number or a multiple of 3 as the quotient, resulting in a remainder other than 8. It's just a no-show!
7: Seven makes a grand entrance! When you divide a three-digit number by 7, adventures begin! It's possible to get a remainder of 8. For example, 259 divided by 7 equals 37 with a remainder of 8! Circus act accomplished!
8: Ah, the magnificent number 8! When you divide a number by itself, you get the quotient as 1. So the remainder will always be 0, not 8. This one is quite a trickster!
9: Last, but not least, the number 9! When dividing a three-digit number by 9, you can indeed get a remainder of 8. For example, 815 divided by 9 equals 90 with a remainder of 8! Applause for number 9!
And there you have it! The divisors for which you can get a remainder of 8 when dividing a three-digit number by a one-digit number are 1, 3, 7, and 9. Keep spreading joy and laughter just like I do, my friend!
To find the divisors for which you can get a remainder of 8 when dividing a 3-digit number, we need to consider all possible divisors and check if they satisfy the given condition.
Let's start by understanding what remainder means. When we divide a number (dividend) by another number (divisor), the remainder is the amount left over after the division process is complete.
To find the divisors for which we get a remainder of 8, we can use the following steps:
1. Consider all the 1-digit numbers (divisors) from 1 to 9 since these are the possible options.
2. Choose a 3-digit number as the dividend.
3. Divide the dividend by each of the 1-digit divisors and calculate the remainder.
4. If the remainder is 8 for any of the divisors, we have found a divisor that satisfies the given condition.
Let's try an example to illustrate this process. Suppose we choose the dividend as 378.
Dividend: 378
Now, we divide 378 by each of the possible divisors from 1 to 9:
- Dividing 378 by 1: Remainder = 0
- Dividing 378 by 2: Remainder = 0
- Dividing 378 by 3: Remainder = 0
- Dividing 378 by 4: Remainder = 2
- Dividing 378 by 5: Remainder = 3
- Dividing 378 by 6: Remainder = 0
- Dividing 378 by 7: Remainder = 0
- Dividing 378 by 8: Remainder = 2
- Dividing 378 by 9: Remainder = 0
From the example above, we can see that when dividing 378 by 4 and 8, we get a remainder of 2. However, we don't have any divisors that give us a remainder of 8.
By repeating the above steps for different 3-digit numbers, you can check if any divisors will give you a remainder of 8. However, based on the nature of division, it is unlikely to find a divisor for which the remainder is 8 consistently for all 3-digit numbers.
To have a remainder of 8, the divisor must be greater than 8. The only single digit greater than 8 is 9, so 9 must be the divisor.
Since 8 is 1 less than 9, the dividend needs to be one less than a multiple of 9. Candidates are values of 3 digits, as the problem states.
The lowest 3-digit number satisfying the condition is 107 (which is 9x12 - 1); the highest is 998 (which is 9x111 - 1).
Some answers:
107 / 9 = 11 R8
125 / 9 = 13 R8
998 / 9 = 110 R8
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