What is the remainder when 21^32 is divided by 50?

Question

A. 1
B. 11
C. 21
D. 31
E. 41

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impeditrem · Answer

We can look for a pattern by calculating the remainders of  21^n  divided by  50  for consecutive values of n 
 mod(50) = divide each expression below by 50 and pass on the remainder to the next and again divide by 50 till the remainder is constant. 
 
  21^1 = 21 = 21 mod{50}  
  21^2 = 441 = 41 mod{50}  
  21^3 = 21^2 * 21 = 41 * 21 = 861 = 11 mod{50}  
  21^4 = 21^3 * 21 = 11 * 21 = 231 =31 mod{50}  
  21^5 = 21^4 * 21 = 31 * 21 = 651 = 1 mod{50}   
 Once we reach a remainder of 1, the cycle repeats. The cycle length is 5, meaning the remainders for  21^1, 21^2, 21^3, 21^4, 21^5  are the same as  21^6, 21^7, 21^8, 21^9, 21^{10} , and so on. 
 Now, we divide our exponent, 32, by the cycle length: 
  32 = (5 * 6) + 2  
 Since the remainder is 2, 21^{32} will have the same remainder as $21^2$, which we calculated as 41.

Dereno · Answer

Any number with unit digit as 1, that is raised to the power of n, will be of the form (a1)^n.

(a1)^n will always yield the unit digit as 1.

But, the tens digit will be (n*a) .

How is this correct : 11^2 = 121. Tens digit = (1*2) and the Unit digit as 1.

Similarly, 11^3 = 1331. The unit digit is 1, and the tens digit is (3*1) =3.

So applying the same logic.

21^32 = (32*2) as tens digit and 1 as unit digit.

So, the last two digits will be 41.

So, when 41 is divided by 50. The remainder is 41 or negative remainder of -9.

Hence,  Option E

ExpertsGlobal5 · Answer

Video explanation:

https://www.youtube.com/watch?v=-k_inBP_wvU

Adit_ · Answer

The remainders go like 21,41,61.

Thus using a+(n-1)d formula you get a value divide by 50 and remainder u get is 41.

Answer:  Option E

pappal · Answer

21^32 % 50=? where a%b means remainder when a is divided be b
 i tried this way
21^2=(20+1)^2=400+1+40 when divided by 50 remainders are 0+1+40 
equation 21^32 % 50 can be expressed as 41^16 % 50-------i
and (40+1)^2=1600+1+80 when divided by 50 remainders are 0+1+80
so i reduces to 81^8 % 50
similarly i finally reduces to 161^4%50 or 321^2%50 or 641%50
641%50=41 
so D

What is the remainder when 21^32 is divided by 50?

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What is the remainder when 21^32 is divided by 50?

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615 to 715 in 15 Days (Ria's Strategies)

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