UAE
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 ^@
- 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. ^@ - 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. ^@ - 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. ^@ - 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. ^@ - Hence, the value of the integer ^@ cba ^@ is ^@ 576. ^@
