The letters ^@ a, b ^@ and ^@ c ^@ stand for non-zero digits. The integer ^@ abc ^@ is a multiple of ^@ 3 ^@ the integer ^@ cbabc ^@ is a multiple of ^@ 15, ^@ and the integer ^@ abcba ^@ is a multiple of ^@ 8. ^@ What is the value of the integer ^@ cba? ^@


Answer:

^@ 576 ^@

Step by Step Explanation:
  1. We know that a number is divisible by ^@ 8 ^@ if it's last ^@ 3 ^@ digits are divisible by ^@ 8. ^@
    Given, ^@ abcba ^@ is a multiple of ^@ 8. ^@
    Therefore ^@ cba ^@ is a multiple of ^@ 8. ^@
  2. Also, ^@ abc ^@ is given to be a multiple of ^@ 3. ^@
    Since the sum of the digits of ^@ abc ^@ and ^@ cba ^@ are the same, ^@ cba ^@ is also a multiple of ^@ 3. ^@
    Therefore, ^@ cba ^@ is a multiple of ^@ 24. ^@
  3. We are given that ^@ cbabc ^@ is a multiple of ^@ 15 ^@ and ^@ c \ne 0 ^@ (given).
    ^@ \implies c = 5 ^@
    Now, ^@ cbabc ^@ is a multiple of ^@ 15 ^@ therefore ^@ cbabc ^@ is a multiple of ^@ 3. ^@
    ^@ \implies ^@ sum of digits of ^@ cbabc ^@ is a multiple of ^@ 3. ^@
    Also, ^@ a + b + c ^@ is a multiple of ^@ 3, ^@ therefore, ^@ c + b ^@ is a multiple of ^@ 3. ^@
  4. The three-digit multiples of ^@ 24 ^@ starting with ^@ 5 , ^@ which are the possible values of ^@ cba ^@ are ^@ 504, 528, 552, ^@ and ^@ 576. ^@
    Out of the above possible values of ^@ cba, ^@ only ^@ 576 ^@ has ^@ c + b ^@ as a multiple of ^@ 3. ^@
  5. Hence, the value of the integer ^@ cba ^@ is ^@ 576. ^@

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