3
votes
2answers
177 views

How many digits are there in $2^{17}\times 3^2\times 5^{14}\times 7 ?$

How many digits are there in $$2^{17}\times 3^2\times 5^{14}\times 7 ?$$ Question added: I agree with the fellow who asked that if one cannot have 2 and 5 in the number above how we will calculate ...
1
vote
0answers
27 views

Non-repeating decimals in 1/n [duplicate]

Here's a conjecture I made: The number of non repeating decimal places in the base-ten representation of the fraction 1/n, where n is an integer, is equal to whichever is higher: the exponent of 2 in ...
2
votes
3answers
239 views

Why irrational implies having an infinite decimal expansion?

Why irrational means having an infinite decimal expansion?
8
votes
2answers
243 views

Computing the last non-zero digit of ${1027 \choose 41}$?

I am working on the following problem: Let $x_n$ be a sequence of positive odd numbers. If $N$ is the number of ordered pairs $(x_1, x_2, x_3, \dots, x_{42})$ such that $$x_1 + x_2 + x_3 + \dots + ...
1
vote
0answers
109 views

Division of large numbers

Euclidean Algorithm Versus Horner Algorithm. I have come across this problem and I do not manage to find a good example for it. For all the numbers I pick there is no loss. Give an example of a ...
4
votes
3answers
49 views

Graph of the last digit of $x^n$ - why is it symmetric when $n$ is even, and not when $n$ is odd?

I have discovered this fact: "The graph of the last digit of $x^n$ (where $x$ is positive) is asymmetrical if $n$ is odd, and symmetrical if $n$ is even." What is the logic behind this? For ...
7
votes
0answers
79 views

Which digit occurs most often?

Is there any method to calculate, which digit occurs most often in the number $$4 \uparrow \uparrow \uparrow \uparrow 4\ ,$$ the fourth Ackermann-number ? Or would it be necessary to calculate the ...
5
votes
3answers
221 views

What is the smallest natural number n?

What is the smallest natural number n for which there is a natural k, such that, the lasts 2012 digit in the representation decimal of $n^k$ are equal to 1? I don't even know how to start with it ... ...
2
votes
0answers
38 views

A four-digit square is of the form $aabb$. What's $a^2+b^2$? [duplicate]

The 5772 Ulpaniada included the following question: Consider a four digit square number (a number which is the square of a whole number).Its digit notation is $aabb$ (the thousands digit is $a$, ...
2
votes
1answer
68 views

Calculating 6 decimal digits of $3^{\sqrt2}$ using a calculator.

How can we calculate $3^{\sqrt2}$ to 6 decimal digits, using only a semi-basic calculator (Which has the square root too) and a pen and paper? I asked this question from my teacher and he ...
1
vote
2answers
44 views

Divisibility by $9$

Suppose we have a natural number $N$ with decimal representation $A_kA_{k-1}\ldots A_0$. How do I prove that if the $\sum\limits_{i=0}^kA_i$ is divisible by $9$ then $N$ is divisible by $9$ too?
1
vote
1answer
220 views

Period of a decimal expansion

Show that if n is a product of m distinct primes, then the period of the decimal expansion of 1/n is the lowest common multiple of the periods of 1/p over all primes p|n. I understand that the above ...
3
votes
1answer
96 views

The number of zeros in the decimal representation of the factorial of 126

How many zeros are in $126!$ ... the result is $34$. But can I calculate it manually? I have seen How many zeroes are in 100! but I don't think it's helpful.
10
votes
1answer
285 views

How many zeroes are there at the end of the sum $1^1 + 2^2 + 3^3 + \cdots+ 100^{100}$?

Find the number of zeroes at the end of the sum $$1^1 + 2^2 + 3^3 + \cdots+ 100^{100}$$ I tried a lot and my answer came 4700 but it was not correct.
4
votes
1answer
138 views

Length of recurrent strings of numbers in the decimal expansion of $1/p$, where $p$ is prime.

Am I right to assume that: all rational numbers have a recurrent sequence in their decimal expansion, and the length of the expansion of $1/p$, where $p$ is prime, is $p-1$ for sufficiently large ...
3
votes
0answers
98 views

Decimal representation and Peano axioms

I really tried to find similar questions but didn't manage to find them. Please, forgive me if this question is a duplicate. I also apologize for my English. So. The question. We're given five Peano ...
0
votes
2answers
89 views

Definition of period of a decimal representation of a number

I need to define the period of a decimal representation of a number!! Thanks in advance!!
3
votes
1answer
57 views

What is the terminology for the non-repeating portion of a rational decimal?

Given a number co-prime with 10, such as thirteen, we can construct a repeating decimal from its reciprocal: $\frac{1}{13}$ = 0.(076923). If we successively divide this number by a factor of 10 (i.e., ...
3
votes
1answer
493 views

Period of repeating decimals

Note: This question is base sensitive. Therefore, assume we have fixed a base $b$. By abuse of terminology, I will still use the word "decimal". This question revolves around the period of repeating ...
0
votes
6answers
229 views

How to multiply decimal with wholenumber?

How Can I multiply x = (0.35)(80) x = 28 steps by step fastest way I am not going to lie, but it is time for me to take a test without using a calculator. Schools have made me worse by giving us a ...
3
votes
3answers
321 views

Characterising reals with terminating decimal expansions

Show that a number has a terminating decimal expansion if and only if, it is rational and when in lowest terms, its denominator is coprime to all primes other than $2$ and $5$. This is an unsolved ...
14
votes
6answers
618 views

Calculate the 146th digit after the decimal point of $ \frac{1}{293} $

The question is: Calculate the 146th digit after the decimal point of $\frac{1}{293}$ 1 / 293 = 0,00341296928.., so e.g., the fifth digit is a 1. We know that 293 is a prime, probably this would ...
64
votes
5answers
8k views

What is special about the numbers 9801, 998001, 99980001 ..?

Just saw this post, and realized that 1/9801 = ...
4
votes
3answers
796 views

The number of ones in a binary representation of an integer

Is there any relation that tells whether the number of ones in a binary representation of an integer is an even or an odd number?
4
votes
4answers
547 views

Are there any other numbers like $0.999\ldots$?

In a manner similar to how the value $1$ can be represented as $0.(9)$ too, are there any other values that exhibit this property when represented in base 10?