A three-digit number ABC is divided by the two-digit number AC. The quotient is 13 with no remainder. What is the number ABC, if C is not equal to 0?
If someone could explain the reasoning on how to get ABC, I would appreciate it.
we need 13*ac = abc
That means 3*c ends in c
Looks like c must be 5
So, now we have
a5*13 = ab5
15*13 = 195
Well, well, well, it seems we have a mathematical conundrum on our hands! Let's put on our thinking caps and solve this riddle together!
So, we have a three-digit number ABC divided by the two-digit number AC, and the quotient is 13 with no remainder. This means that AC can go evenly into ABC exactly 13 times.
Now, let's break it down step by step. Since the quotient is 13, let's assume that A is the same digit in both ABC and AC. So, we have A in the hundreds place for both numbers.
Now, ABC divided by AC gives us 13. We also know that C is not equal to 0. This means that A is multiplied by 10, leaving us with 10A in the tens place.
Now, our final digit B can be anything from 1 to 9, as long as it satisfies the condition that AC can be divided into ABC 13 times. We need to find a number B that makes this equation true.
To do that, we can start by looking at the lowest possible value for ABC, which would be 100A + 10A + 1. Divide this by AC, and if it equals 13, we've found our answer!
So, 110A + 1 divided by AC should give us 13. If we plug in different values for A in this equation, we'll eventually find the correct answer.
I hope this explanation helps you in your quest to find the elusive number ABC! Math can be quite the clown sometimes, but with a bit of patience and a lot of laughter, you'll crack the code in no time!
To find the number ABC, we need to consider the division of a three-digit number by a two-digit number, where the quotient is 13 with no remainder.
Let's break down the three-digit number ABC into its respective place values, where A represents the hundreds place, B represents the tens place, and C represents the ones place.
Now, we know that the division of ABC by AC yields a quotient of 13 with no remainder. Since the quotient is 13, it implies that the hundreds place digit, A, is at least equal to 1 (because 13 is greater than 9, the largest single-digit number).
Since the dividend has three digits, and the divisor has two digits, we can infer that A must be greater than or equal to 1. But since the quotient is a two-digit number (13) and C is not equal to 0, it implies that A must be equal to 1, because otherwise, the quotient would be larger than 13.
So, we conclude that A = 1. Now, let's consider the remaining two digits, B and C.
With A = 1, the three-digit number ABC becomes 1BC. We need to find the appropriate values for B and C.
Dividing 1BC by 1C (where A = 1), we get a quotient of 13 with no remainder:
_________
1C | 1BC
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-
-
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1
1C
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0
From this division, we can see that the remainder is 0. Since the remainder is 0, it implies that the last digit of 1BC (which represents the ones place) is divisible by C. In other words, C is a factor of B.
We also know that the quotient is 13, so C must be a number less than or equal to 9 (since the largest two-digit number divisible by a two-digit number is 99). Therefore, we can conclude that C must be a factor of B between 1 and 9.
So, there are several pairs of possible values for C and B that satisfy these conditions, such as:
C = 1, B = 1
C = 2, B = 2
C = 3, B = 6
C = 4, B = 4
C = 6, B = 3
C = 8, B = 8
C = 9, B = 9
Therefore, the possible values for ABC are 111, 222, 336, 444, 663, 888, and 999.
To find the three-digit number ABC, we need to understand how division works.
Let's break down the problem step by step:
1. We are given that the three-digit number ABC is divided by the two-digit number AC. This means that AC is the divisor.
2. We are also given that the quotient (the result of the division) is 13 with no remainder. This means that the three-digit number is a multiple of 13.
Now, let's think about the possible values for A, B, and C:
1. A: Since ABC is a three-digit number, A cannot be 0. If A were 0, the number would have two digits.
2. C: We are given that C is not equal to 0. This implies that C must have a non-zero value.
3. B: Since the quotient is 13 with no remainder, we know that B must be the remainder of the division (in this case, 0). When a number is divided by 13, if there is no remainder, then the number is already a multiple of 13.
Putting it all together:
Since B is the remainder of the division and the quotient is 13, we can determine that B must be 0. Therefore, ABC is a multiple of 13 and ends with 0.
Now we need to find the acceptable values for A and C:
A can be any value from 1 to 9 (excluding 0), as A represents the hundreds place (since ABC is a three-digit number).
C can be any value from 1 to 9 (excluding 0), as C represents the units place (since C is not equal to 0).
By combining the possible values for A and C, we can find all the three-digit numbers ABC that meet the given conditions.