Now have a read of this article.
How to Find the Last Digits of a Positive Power of Two - Exploring Binary
Please someone tell me how to work out the TENS digit of the answer of WITHOUT using calculator.
I believe that it is somehow related to the principles such as
the original question also included the units digit but it should always be 6.
Thanks in advance.
Now have a read of this article.
How to Find the Last Digits of a Positive Power of Two - Exploring Binary
A few calculations (you should be able to multiply these easily w/o the abacus)
016 = 16^1
x16
===
_56 = 16^2 (16 times 16 = 256, but we only need the tens & ones digits)
x16
===
_96 = 16^3
x16
===
_36 = 16^4
x16
===
_76 = 16^5
x16
===
_16 = 16^6 (same as 16^1, exponent difference is 5)
298/5 = 59.9
59*5 = 295
1+5=6
1+295=296
&(add 2 to the exponents)
see prior calculation above for the tens & ones digits for which is the same as
.
Note that
Thus
Note that (easily done mentally)
Thus
Thus
Note that
You easily find out that
Thus,
We get that
Note that .
You easily find out that .
Thus, .
Note that .
You easily find out that .
Thus, .
You easily find out that .
Thus, .
Conclusion : the tens digit of is .
I know my method is long, boring and repetitive, but eh ? it gets the job done
I mean the before-last digit, of course ... not the 10th digit from the left
67259685376506838351825694125205607357734081036908 94500677045631406539332516306882577142397357830536 43998864024685999522991598722297525821288734761044 09052127576185436156214378455780845781465327759636 97038693955559119408307906472347614286627358580697 16218116239821639793538226085261772687392217674908 12009984323096665122511297763735177027009515287152 894672896
I was talking about the red digit, (I believe it is the question) ...
Hello, ukorov!
My method is similar to Aiden's . . .
Note the last two digits of consecutive powers of 16 . . .Find the tens digit of the answer of without using calculator.
. .
We find that the last two digits have a 5-digit cycle: .
. . Hence: .
Since
. .
Therefore, the tens digit of is
Edit: corrected typo . . . Thanks, Galactus!