A two-digit number ^@ ab ^@ is multiplied by its reverse ^@ ba ^@. The ones (units) and tens digits of the four-digit resultant number are both ^@ 0. ^@ What is the value of the smallest such two-digit number ^@ ab ?^@


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Answer:

^@ 25 ^@

Step by Step Explanation:
  1. Since the units digit of the resultant number is zero, either ^@ a ^@ or ^@ b ^@ must be ^@ 5. ^@
    Without loss of generality, assume ^@ a = 5. ^@
    Therefore, ^@ b ^@ is even.
  2. Since the answer ends in ^@ 00 ^@
    ^@ \implies ^@ The answer is a multiple of ^@ 100 ^@ and hence is a multiple of ^@ 25. ^@
    Since ^@ b \ne 0 ^@ and ^@ ba ^@ ends in ^@ 5, ba ^@ is a multiple of ^@ 25. ^@
    The only ^@2^@-digit multiples of ^@25^@ ending in ^@5^@ are ^@25^@ and ^@75.^@
    From step 1, ^@ b ^@ is even and ^@ 7 ^@ is not an even number.
    Therefore ^@ ba = 25 ^@
    Therefore, the two possible values for ^@ ab ^@ are ^@ 25 ^@ and ^@ 52^@.
  3. Hence, the smallest value of the two-digit number ^@ ab ^@ is ^@ 25. ^@

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