If x= 1/(2^2*3^2*4^2*5^2) is expressed as a decimal, how many distinct

You don't really need to do long division here if you don't want to, as long as you know that 1/9 = 0.11111....., and 2/9 = 0.22222...., and so on.

If we multiply a number by 10, or by 100, or by any other power of 10, we're just moving the decimal point over. We won't change the number of distinct nonzero digits we have in our number. So we can freely multiply the fraction by any power of 10 without changing the answer. We can start by multiplying the fraction by 10^2 = 2^2 * 5^2, to get:

1/3^2*4^2

which is already a bit simpler. But we can keep going - we can multiply by 100^2 = (4*25)^2 = 4^2 * 25^2 to get the 4s out of the denominator as well. So this fraction

25^2/3^2 = 625/9

has the same number of distinct digits as the fraction in the question. Since 630/9 = 70, then 621/9 = 69, so 625/9 = 69 + 4/9 = 69.44444..... which has three distinct digits.

But I think most people would just do long division, either with 1/144, or with 1/14,400, which will also work, and probably is just as fast. I hate doing long division so I always look for another way!

\(\frac{1}{(2^2*3^2*4^2*5^2)}\) = \(\frac{1}{2^2} * \frac{1}{3^2} * \frac{1}{4^2} * \frac{1}{5^2}= \frac{1}{100} * \frac{1}{9} * \frac{1}{16}\)

= \((.01 * .0625) * \frac{1}{9}\) (division by 9 leads to a non-terminating decimal so leave it for the end)

= .000625/9

= .0000694444...

So we have 3 distinct non zero digits.

Answer (C)

Similar questions to practice:
https://gmatclub.com/forum/if-3-4-2-3-5- ... 98180.html
https://gmatclub.com/forum/if-d-1-2-3-5- ... 44440.html
https://gmatclub.com/forum/if-t-1-2-9-5- ... 29447.html
https://gmatclub.com/forum/if-1-2-11-5-1 ... 23889.html

Hope it helps.
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we try to make dominator the multiple of 10,
so we multiple with 5^4

we have 5^4/9, we do not care how many 10 factors are there is the dominator

divide
694444444

so, there is only 3 distict

tricky

Following is the simple calculation.

Posted from my mobile device

Took 01:02.

1st) do some fast math in your head: 4*9*16*25 = 64 * 225 = 10* 225 + 50* 225 + 4 * 225 = 2250 + 50*200 + 50*25 + 900 = 12250 + 1250 + 900 = 14 400 (or: 100*16*10 - 1600 = 16000 - 1600 = 14400)
2nd) switch the fraction into a decimal
done

the easy way.

I used indices to solve this question.

Multiplying the denominator;

x = \(\frac{1}{2^2*3^2*4^2*5^2}\)

x = 1/120^2

(i got 120 by multiplying 2*3*4*5)

x = \(\frac{1}{(12*10)^2}\)

x = \(\frac{1}{144*100}\)

x = \(\frac{1}{14400}\)

three distinct numbers!

Let me know if this method is sound. Thanks

Give me kudos if you like my explanation

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