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

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

Quote:
Now, all these numbers ending with 1, when divided by 5 give result ending in XXXX.2

What if instead of five there were a different number.

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

Thank you.
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nonameee wrote:
Bunuel, the first part is systematic. The second part is kind of fuzzy:

Quote:
Now, all these numbers ending with 1, when divided by 5 give result ending in XXXX.2

What if instead of five there were a different number.

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

Thank you.

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.
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Re: (3^32)/50 gives remainder and {.} denotes fractional part of [#permalink]
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.
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Re: (3^32)/50 gives remainder and {.} denotes fractional part of [#permalink]
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BN1989 wrote:
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.

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=0.02;
21/50=0.42;
1001/50=20.02;
...
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Re: (3^32)/50 gives remainder and {.} denotes fractional part of [#permalink]
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
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Re: (3^32)/50 gives remainder and {.} denotes fractional part of [#permalink]
Galiya wrote:
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

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
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Re: (3^32)/50 gives remainder and {.} denotes fractional part of [#permalink]
virtualanimosity wrote:
(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

$$= 3^{32}$$

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

$$3^{4} =81 =50 +31$$

$$= (M50 + 31)^{8}$$

$$= 31^{8}$$

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
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(3^32)/50 gives remainder and {.} denotes fractional part of [#permalink]
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virtualanimosity wrote:
(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

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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Re: (3^32)/50 gives remainder and {.} denotes fractional part of [#permalink]
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Re: (3^32)/50 gives remainder and {.} denotes fractional part of [#permalink]
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