Suppose 1 ≤ x, y, z ≤ 9 are integers such that x > z. Let α and β be the three-digit numbers given by α = xyz and β = zyx so that x, y, z are the digits as they are shown. Set γ = α − β. How many of the following statements are always true? Select the correct answer.
3
To solve the problem, we need to analyze each statement by calculating the difference \(\gamma = \alpha - \beta\) where:
\(\alpha = 100x + 10y + z\)
\(\beta = 100z + 10y + x\)
To find \(\gamma\), compute:
<[\( \gamma = (100x + 10y + z) - (100z + 10y + x) = 99x - 99z = 99(x - z) \)\]
Since \(\gamma = 99(x - z)\) where \(x > z\) and 1 ≤ \(x, y, z ≤ 9\), let's analyze each statement:
- 2 does not divide \(\gamma\).
Since 99 is not divisible by 2, \(\gamma\) is always odd. Thus, 2 never divides \(\gamma\). - 3 divides \(\gamma\).
99 is divisible by 3. Therefore, \(\gamma\), which is \(99(x-z)\), is always divisible by 3. - 5 does not divide \(\gamma\).
99 is not divisible by 5, and since \(99(x-z)\) never results in a multiple of 5 for integer \(x\)\) and \(z\) that differ by less than 5, 5 never divides \(\gamma\). - 7 does not divide \(\gamma\).
99 is not divisible by 7, so \(99(x-z)\) will not be divisible by 7, since x and z differ only by a few digits allowed by the problem. - 11 divides \(\gamma\).
99 is divisible by 11. Therefore, \(\gamma\) is always divisible by 11. - 13 does not divide \(\gamma\).
Similar reasoning to the previous non-divisible checks: 99 is not divisible by 13, making it impossible for \(99(x-z)\) to be divisible by 13.
Based on the analysis above, the statements that are always true are 2, 3, and 5. Hence, the number of statements that are always true is:
- Statement 1: True
- Statement 2: True
- Statement 3: True
- Statement 4: True
- Statement 5: True
- Statement 6: True
The correct option is 3 as only three statements are always true.
Important Questions from Number System
Consider the following statements :
1. (25)! + 1 is divisible by 26
2. (6)! + 1 is divisible by 7
Which of the above statements is/are correct ?
If the sum S is divided by 8, what is the remainder ?
If the sum S is divided by 60, what is the remainder ?
Find the sum of squares of the greatest value and the smallest value of K in the number so that the number 45082K is divisible by 3.
How many composite numbers are there from 53 to 97 ?
