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Suppose 1 ≤ x, y, z ≤ 9 are integers such that x > z. Let a and ß be the three-digit numbers given by a = xyz and ß = zyx so that x, y, z are the digits as they are shown. Set y = a - B. How many of the following statements are always true?
I. 2 does not divide Y.
II. 3 divides y.
III. 5 does not divide y.
IV. 7 does not divide y.
V. 11 divides y.
VI. 13 does not divide y.
Select the correct answer.
Explanation
Given that 1 ≤ x, y, z ≤ 9 and x > z. The three-digit numbers are:
a = 100x + 10y + z
ß = 100z + 10y + x
The difference is given by y = a - ß:
y = (100x + 10y + z) - (100z + 10y + x) = 99x - 99z = 99(x - z).
Let k = x - z. Since 1 ≤ z < x ≤ 9, k can be any integer from {1, 2, 3, 4, 5, 6, 7, 8}.
So, y = 99k = 32 × 11 × k.
- I. 2 does not divide y: False. If k = 2, y = 198, which is divisible by 2.
- II. 3 divides y: Always True. y = 3 × (33k).
- III. 5 does not divide y: False. If k = 5, y = 495, which is divisible by 5.
- IV. 7 does not divide y: False. If k = 7, y = 693, which is divisible by 7.
- V. 11 divides y: Always True. y = 11 × (9k).
- VI. 13 does not divide y: Always True. Since 13 is prime and does not divide 99 or any k ∈ {1, ..., 8}.
Statements II, V, and VI are always true. Total = 3.
SIMILAR QUESTIONS
Suppose x, y, z are three positive integers such that x< y<>z and xyz = 7 2. Which one of the following values of S yields more than one solution to the equation x + y + z-S?
Suppose x is the smallest integer greater than 3 such that when it is divided by 6 or 8, the remainder is 3. y is the smallest integer greater than 2 such that when it is divided by 6 or 8, the remainder is 2. What is x - y ?