If x and y are positive integers

Question

If x and y are positive integers, the units digit of  x^{43}+y^{82}  must be identical to the units digit of which of the following expressions?

(A)  x^{12}+y^{16}  
 (B)    x^{15}+y^{24}  
 (C)    x^{23}+y^{46}   
 (D)    x^{33}+y^{60}   
 (E)    x^{41}+y^{81}

Daphnee · Answer

Hint: To solve this, we use a specific mathematical property: For any integer raised to a power, its units digit repeats in a cycle of 4. Therefore, we just need to divide the exponent by 4 and look at the remainder to determine if the units digits match

onlyPlanA · Answer

In the equation:   x^{43}+y^{82} , when the exponents of x and y are divided by the cyclicity 4, the remainders for the exponent of x is 3 and for the exponent of y is 2.

Of all the options, only C has exponents that, when divided by 4, gives the same remainders. 
  So, C is the answer.

modiassumenda · Answer

Solution: 
To have the same units digit, the exponents must have the same remainder when divided by 4.
For (x^{43}):
 
 (43 \div 4) leaves remainder 3.  
For (y^{82}):
 
 (82 \div 4) leaves remainder 2.  
Now check each option:
 
  (A)  (12 = 0), (16 = 0) ❌ 
  (B)  (15 = 3), (24 = 0) ❌ 
  (C)  (23 = 3), (46 = 2) ✅ 
  (D)  (33 = 1), (60 = 0) ❌ 
  (E)  (41 = 1), (81 = 1) ❌  
Only  Option (C)  has the same remainders as the original exponents.
 Answer: (C) (x^{23}+y^{46})

Docmba68 · Answer

Bunuel  
Can you explain why in this question we have to take cyclicity of 4 although no particular number has been mentioned which has cyclicity of 4.

Bunuel · Answer

Yes, some units digits have shorter cycles.

For example:

0, 1, 5, and 6 have cycle 1.
4 and 9 have cycle 2.
2, 3, 7, and 8 have cycle 4.

But cycle 4 still works for all of them, because 4 is a multiple of both 1 and 2.

For example, 9 has cycle 2: 9, 1, 9, 1. But after 4 powers, the same pattern also repeats. So using cycle 4 is still valid.

Since x and y are unknown, we use cycle 4 because it covers every possible units digit.

If x and y are positive integers

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If x and y are positive integers

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