A two-digit number abab is multiplied by its reverse baba. The ones (units) and tens digits of the four-digit resultant number are both 0.0. What is the value of the smallest such two-digit number ab?ab?


Answer:

2525

Step by Step Explanation:
  1. Since the units digit of the resultant number is zero, either aa or bb must be 5.5.
    Without loss of generality, assume a=5.a=5.
    Therefore, bb is even.
  2. Since the answer ends in 0000
    The answer is a multiple of 100100 and hence is a multiple of 25.25.
    Since b0b0 and baba ends in 5,ba5,ba is a multiple of 25.25.
    The only 22-digit multiples of 2525 ending in 55 are 2525 and 75.75.
    From step 1, bb is even and 77 is not an even number.
    Therefore ba=25ba=25
    Therefore, the two possible values for abab are 2525 and 5252.
  3. Hence, the smallest value of the two-digit number abab is 25.25.

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