Question: A symmetric number of an another one is a number where the digit are reversed. For instance, 123 is the symmetric of one of 321. Thus the difference of a number and its symmetrical must be divisible by which of the following?
- 4
- 5
- 6
- 7
- 9
Correct Answer: E
Solution and Explanation:
Approach Solution 1:
Given: A symmetric number of an another one is a number where the digit is reversed.
Example: 123 is the symmetric of one of 321
Candidates need to find out: The difference between a number and its symmetrical must be divisible by which number.
Let us consider the example of three-digit symmetric numbers {abc} and {cba}. Three digit numbers can be represented as: {abc}=100a+10b+c and {cba}=100c+10b+a. The difference would be:
{abc}-{cba}=100a+10b+c-(100c+10b+a)
=99a-99c=99(a-c).
Two digits: {ab} and {ba}. {ab}-{ba}=10a+b-(10b+a)=9a-9b=9(a-b)
Hence the difference of two symmetric numbers (2 digit, 3 digit, ...) will always be divisible by 9.
Approach Solution 2:
The problem statement states that:
Given: A symmetric number of an another one is a number where the digit is reversed.
Example: 123 is the symmetric of one of 321
Asked: Find out the number divisible by the difference between a number and its symmetrical.
This is a different process for approach.
We can also try plugging in some numbers.
2 digits: 34 and 43 -> the difference is 9, which is divisible by 9
3 digits: 123 and 321 -> the difference is 198, which is divisible by 9
3 digits: 111 and 111 -> the difference is 0, which is divisible by 9
All the random numbers show a difference of 9.
Hence the difference of two symmetric numbers will always be divisible by 9.
Approach Solution 3:
The problem statement states that:
Given: A symmetric number of an another one is a number where the digit is reversed.
Example: 123 is the symmetric of one of 321
Asked: Find out the number divisible by the difference between a number and its symmetrical.
The answer should be 9 because the difference will always be a multiple of 9.
For instance,
xyz = 100x+10y+z
zyx= 100z+10y+x
(zyx) - (xyz) = 99(z-x)
which is always divisible by 9 or 11.
Hence the difference of two symmetric numbers will always be divisible by 9.
“A symmetric number of an another one is a number”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “Advanced GMAT Quant”. The candidate must have a solid knowledge of mathematics to solve GMAT Problem Solving questions. The candidates can go through GMAT Quant practice papers to analyse multiple questions to strengthen their mathematical skills.
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