If x is an integer greater than 1, what is the remainder when x^4 is

Question

If x is an integer greater than 1, what is the remainder when x^4 is divided by 10?

(1) x is not a multiple of 5.
(2) x is a multiple of 2.

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crimson_king · Answer

The answer for this should be C.

If we look at option A, if the number is not a multiple of 5 then it can be a multiple of 2,3 or 7

If it is a multiple of 2 that is not divisible by 5, the remainder will always be 6 if divided by 10 as all the multiples of 2 not divisible by 5 when raised to the power of 4 will end with the unit's digit being 6 & the remainder will be 6 when divided by 10.

If it's a multiple of 3 or 7, the unit's digit will be 1 & the remainder will hence be 1 when divided by 10

Since we cannot determine the remainder with definite proof, (A) is incorrect & by default (D) is also incorrect.

If we only consider (B), we saw that the multiples of 2 that are not divisible by 5 will give the value of the remainder as 6. But 10 is also a multiple of 2 & the remainder will be 0 in this case. So (B) alone is also not sufficient.

If we combine both (A) & (B), then we get a clear answer as the multiples of 10 will not be considered & we will always have our remainder as 6. The correct option hence is (C)

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Shubhradeep · Answer

Answer Option C should be Correct

If any number is divided by 10, the remainder is the unit's digit. First Let us see what are the unit's digits for different cases of unit
digits of a number.

1^4=1
2^4=6
3^4=1
4^4=6
5^4=5
6^4=6
7^4=1
8^4=6
9^4=1

For Statement (1) X>1 and x not a multiple of 5
 
   Unit's Digit    Unit's digit after ^4   
  1  1 ❌   
  2  6  
  3  1  
  4  6  
  5  5 ❌  
  6  6  
  7  1  
  8  6  
  9  1  
  0  0 ❌

Now we have 2 different cases of unit's digit (1 and 6); so can't reach to a conclusion.  Hence, not sufficient.

For statement (2) x>1 and x is an even number
 
   Unit's Digit    Unit's digit after ^4   
  1  1 ❌  
  2  6  
  3  1 ❌  
  4  6  
  5  5 ❌  
  6  6  
  7  1 ❌  
  8  6  
  9  1 ❌  
  0  0

The leftovers, all of them have unit's digit of 6 except 0. So can't reach conclusion.  Hence, statement
2 is not Sufficient

If 1 and 2 are clubbed, we get
 
   Unit's Digit    Unit's digit after ^4   
  1  1 ❌  
  2  6  
  3  1 ❌  
  4  6  
  5  5 ❌  
  6  6  
  7  1 ❌  
  8  6  
  9  1 ❌  
  0  0 ❌

So only remainder is 6.

Correct option should be C

If x is an integer greater than 1, what is the remainder when x^4 is

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Unit's Digit	Unit's digit after ^4
1	1 ❌
2	6
3	1
4	6
5	5 ❌
6	6
7	1
8	6
9	1
0	0 ❌