Question

Write the unit digit of the cube of 1243525

Answer 1

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♦ Elementary Number Theory ♦

→ Recall : ( 1243525 ) can be written as [ 1243520 + 5 ]

 • Note : ( a + b )³  = a³ + b³ + 3ab( a + b )
  => Every term of the expansion is divisible by 'a' , and all the factors of 'a' as well 

  • This is common in case of every power of the Binomial expansion
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 => ( 1243525 )³ = ( 1243520 + 5 )³
                           = 10( ... ) + 5³
                           = 10( ... ) + 125
                           = 10( ... + 12 ) + 5

 => Unit Digit of ( 1243525 )³ is 5 
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 ♦ Cyclicity :
     • Any number with unit's digit : '0', '1', '5' , '6' , raised to any power, will inherit the same unit's digit 

 => Unit Digit of ( 1243525 )³ is 5 
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♦ Modular Arithmetic :
   • ( 1243525 ) ≡ 5 ( mod 10 )
=> ( 1243525 )³ ≡ ( 5 )³ = 125 ≡ 5 ( mod 10 )
=> Unit Digit of ( 1243525 )³ is 5  

Note : We utilize the concept of Remainders and that the Unit's Digit is the Remainder when the whole Number is divided by '10' 
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♦ SPAM ANSWER :
  → ( 1243525 )³ = 1922930387125328125

 => Unit's digit = '5'
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^_^ Hope it helps