(3^32)/50 gives remainder and {.} denotes fractional part of

Question

(3^32)/50 gives remainder and {.} denotes fractional part of that.The fractional part is of the form (0.bx). The value of x could be
1. 2
2. 4
3. 6

kalpeshchopada7 · Accepted Answer

powers of 3 give same last digit at specific intervals. which is:
Power - last digit
1 - 3
2 - 9
3 - 7
4 - 1

Hence, 32nd power of 3 will also be ending with 1. Now, all these numbers ending with 1, when divided by 5 give result ending in XXXX.2, and hence when divided by 50 should result in xxx.X2. Hence the value of x in the question should be 2. OA -1

Is it right?

GMAT TIGER · Answer

Good work. Thats 2.

nonameee · Answer

Bunuel, the first part is systematic. The second part is kind of fuzzy:

What if instead of five there were a different number.

Is there a more systematic approach to these kind of problems?

Thank you.

Bunuel · Answer

Solution above is pretty systematic.

If you are interested in the easiest approach for different numbers then it'll depend on that numbers and answer choices.

BN1989 · Answer

Can anybody explain to me why 1 was divided by 5? If I divide a number that ends with 1 by 50, all kinds of different remainders and therefore digits could result.

Bunuel · Answer

That's not true. ANY number with 1 as the units digit when divided by 50 gives 2 as the last digit after decimal point:
1/50=0.0 2 ;
21/50=0.4 2 ;
1001/50=20.0 2 ;
...

galiya · Answer

for such kind of problems how to realize from what power of an integer should i start looking for the pattern?
since, if start listing the powers of three from 0, ill get the unit digit of 3^32=...7
3^0=1
3^1=3
3^2=9
3^3=27
...
suppose its possible to face with a problem when it'll be crucial to know this

Bunuel · Answer

When looking for a pattern of the last digit of a^n ALWAYS start from n=1 (otherwise if you start from n=0 then all integers will have 1 as the first number in pattern). For more check: math-number-theory-88376.html

mindmind · Answer

= 3^{32}

= (3^{4})^{8}

3^{4} =81 =50 +31

= (M50 + 31)^{8}

= 31^{8}

Last digit 1

For division by 10 & multiple thereof ... find the last digit

So remainder is 1/50

i.e 0.02 --- 0.bx

x=2

Option 1

TeamGMATIFY · Answer

This question tests your knowledge about cyclicity of the numbers.
Cyclicity: The power after which the units digit of a number repeats itself

Eg: 2^1  = 2,  2^2 = 4,  2^3  = 8,  2^4  = 16, 2^5  = 32
The units digit repeats after 4 powers hence cyclicity = 4
2, 3, 7, 8: Cyclicity 4
5, 6: Cyclicity 1
4,9: Cyclicity 2

3^{32}  will end with the units digit 1
 3^{32}  will be of the form ----1
Therefore  3^{32} /50 will be of the form ----.b2
Hence x = 2

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(3^32)/50 gives remainder and {.} denotes fractional part of

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(3^32)/50 gives remainder and {.} denotes fractional part of

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