Question

Consider a 3-digit integer x with distinct digits. Let y be the integer formed by swapping
x's units and hundreds digits. What is the least prime number that perfectly divides the
absolute difference between x and y?​

Answer 1

Question :- Consider a 3-digit integer x with distinct digits. Let y be the integer formed by swapping x's units and hundreds digits. What is the least prime number that perfectly divides the absolute difference between x and y ?

Solution :-

Let us assume that, 3 digit integer x is abc , where ,

• unit digit = c
• tens digit = b
• hundreds digit = a .

so,

→ x = (100a + 10b + c)

now, we have given that, y is formed by swapping unit digits and hundreds digits of x .

then,

• unit digit = a
• tens digit = b
• hundreds digit = c

so,

→ y = (100c + 10b + a)

therefore,

→ x - y = (100a + 10b + c) - (100c + 10b + a)

→ x - y = 100a - a + 10b - 10b + c - 100c

→ x - y = 99a - 99c

→ x - y = 99(a - c)

→ x - y = 3 * 3 * 11 * (a - c) .

Hence, we can conclude that, the least prime number that perfectly divides the absolute difference between x and y is 3 . {when value of (a - c) is not equal to 2.}

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Answer 2

Given : 3-digit integer x with distinct digits.

y be the integer formed by swapping x's units and hundreds digits.

To Find : the least prime number  that perfectly divides the absolute difference between x and y

Solution:

3-digit integer x with distinct digits.   =  abc

100a + 10b + c

y be the integer formed by swapping x's units and hundreds digits.

=> cba

100c + 10b + a

Difference  =  100a + 10b + c - (100c + 10b + a)

= 99a  - 99c

= 99(a - c)

= 3 * 3 *11 (a - c)

3 is the  least prime number  that perfectly divides the absolute difference between x and y  

and 2 can b least if | a - c |  = 2.

11  is the greatest prime number that perfectly divides the absolute difference between x and y

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