The three digit number XYZ when divided by 8, gives as quotient the twoYZ when divided by 8, gives as quotient the two digit number Zxremainder Y. The number XYZ is
Let's break down the information given step by step.
First, we are told that XYZ divided by 8 gives the quotient twoYZ. This means that:
XYZ/8 = 2YZ
Next, we are told that twoYZ divided by 8 gives the quotient Zxremainder Y. This means that:
2YZ/8 = Zxremainder Y
To simplify this, we can write it as:
YZ/4 = Zxremainder Y
Since Y is in the ones place and Z is in the tens place, we can write this equation as:
(10Y + Z)/4 = Z(10) + Y
Simplifying further, we get:
10Y + Z = 4Z(10) + 4Y
Expanding, we get:
10Y + Z = 40Z + 4Y
Combining like terms, we get:
6Y + Z = 40Z
Since Y and Z are both positive integers, we know that Y and Z must both be less than or equal to 9. Therefore, Z can only equal 1.
Substituting Z = 1 into the equation, we get:
6Y + 1 = 40(1)
Simplifying, we get:
6Y + 1 = 40
Subtracting 1 from both sides, we get:
6Y = 39
Dividing both sides by 6, we get:
Y = 39/6
Since Y must be a positive integer, we find that Y = 6.
Therefore, the three-digit number XYZ is 16Y, or 161.
To solve this problem, let's break it down step-by-step:
Step 1: Let's represent the three-digit number XYZ as 100X + 10Y + Z. This is because the hundreds digit is represented by the value of X, the tens digit is represented by the value of Y, and the units digit is represented by the value of Z.
Step 2: The problem states that when XYZ is divided by 8, the quotient is the two-digit number twoYZ. Therefore, we can write the equation:
100X + 10Y + Z = 8(20Y + Z)
Step 3: Expanding the equation:
100X + 10Y + Z = 160Y + 8Z
Step 4: Simplifying the equation:
100X - 160Y + 10Y - 8Z + Z = 0
Step 5: Combining like terms:
100X - 150Y - 7Z = 0
Step 6: To determine the possible values of X, Y, and Z, we can eliminate the variables one by one.
- From the equation, we can see that Z must be a multiple of 10, since dividing it by 8 should result in a two-digit number. Therefore, possible values for Z are 0, 10, 20, ..., 90.
- Let's plug in these values of Z back into the equation to find the possible values of X and Y.
For Z = 0: 100X - 150Y - 7(0) = 0
100X - 150Y = 0
This equation has multiple solutions, so we will try other values of Z.
For Z = 10: 100X - 150Y - 7(10) = 0
100X - 150Y - 70 = 0
100X - 150Y = 70
For Z = 20: 100X - 150Y - 7(20) = 0
100X - 150Y - 140 = 0
100X - 150Y = 140
Continuing this process for Z = 30, 40, 50, 60, 70, 80, and 90.
Step 7: After solving all the equations, we find that the only solution satisfying all the conditions is:
Z = 40
Y = 4
X = 6
Therefore, the number XYZ is 640.