Understanding the Problem: 3-Digit Number Difference
We are given a 3-digit number, represented as XYZ. This means the number can be written algebraically using its digits:
- \(XYZ = 100X + 10Y + Z\)
We are also given its reverse, ZYX. This number can be written as:
- \(ZYX = 100Z + 10Y + X\)
Here, X, Y, and Z represent the digits of the number. X is the hundreds digit, Y is the tens digit, and Z is the units digit. For XYZ to be a 3-digit number, X must be non-zero (\(X \in \{1, 2, ..., 9\}\)). Z can be any digit (\(Z \in \{0, 1, ..., 9\}\)). Similarly, for ZYX to be considered, Z is the hundreds digit and X is the units digit.
The problem defines D as the difference between XYZ and ZYX.
Calculating the Difference (D)
Let's calculate D using the algebraic representations:
- \(D = XYZ - ZYX\)
- \(D = (100X + 10Y + Z) - (100Z + 10Y + X)\)
Now, we simplify the expression by combining like terms:
- \(D = 100X + 10Y + Z - 100Z - 10Y - X\)
- Group the terms involving X, Y, and Z:
- \(D = (100X - X) + (10Y - 10Y) + (Z - 100Z)\)
- \(D = 99X + 0 - 99Z\)
- \(D = 99X - 99Z\)
We can factor out 99 from the expression:
- \(D = 99(X - Z)\)
Finding the Remainder when D is Divided by 99
The question asks for the remainder when D is divided by 99. We have found that \(D = 99(X - Z)\).
Let's perform the division:
- \(\frac{D}{99} = \frac{99(X - Z)}{99}\)
Simplifying this gives:
- \(\frac{D}{99} = X - Z\)
Since X and Z are digits, their difference (\(X - Z\)) is an integer. For example, if X=7 and Z=3, \(X-Z = 4\). If X=2 and Z=8, \(X-Z = -6\). The division \(\frac{99(X - Z)}{99}\) results in the integer \((X - Z)\) with no fractional part.
In mathematical terms, when a number is expressed as \(99 \times k\) (where k is an integer, in our case \(k = X-Z\)), dividing it by 99 always results in the integer k, and the remainder is 0.
Therefore, the remainder when D is divided by 99 is 0.
Similar Questions
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Select the answer using the code given below.
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Important Questions from Number System
What is the value of 1 2 + 2 2 + 3 2 + ......21 2 ?
Which sequence is correct to represent the hierarchical chain of number system?
(Where N - Natural Numbers
W - Whole Numbers
Q - Rational Numbers
Z - Integers)
What must be added to 45680 to make it exactly divisible by 9?
How many zeroes are there at the end of the following product?
1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60
Let XYZ be a three-digit number, where (x + y + Z) is not a multiple of 3. Then (XYZ + YZX + ZXY) is not divisible by
