# Tagged Questions

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### Count numbers with prime digit

Given a number N I need to find the count of the numbers that have atleast one prime digit (2,3,5 or 7) in it. Now N can be upto 10^18.What is the best approach to solve this problem. Example : Let ...
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### Why is Euler's totient function equal to (p-1)(q-1) when N=pq and p and q are prime? [duplicate]

Why is Euler's totient function equal to $(p-1)(q-1)$ when $N=pq$ and $p$ and $q$ are prime? I had my own proof for it but I really don't like it (it feels not intuitive at all because it requires ...
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### Estimations for the number of prime factors, counted with multiplicity (elementary combinatorics)

If $N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid$, where $\Omega(n)$ is the number of prime factors (counted with multiplicity) in $n$, I am trying to reason a crude under-estimate for large $k$ and ...
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### How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $2^{30}3^{13}5^5 7^3$. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
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### parity of powers of prime factors

lets consider the prime factorisation of a number N let the powers of the primes in this factorisation be a,b,c ....and so on. Is there a way to determine whether the number of powers that are even ...
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### Is it true that $p_{n}+p_{n+1}>p_{n+2}$ for all $n\geq 2\ ?$

Let $p_{n}$ denotes the $n$-th prime number. Is it true that $p_{n}+p_{n+1}>p_{n+2}$ for all $n\geq 2\ ?$
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### Bounding the density of finite coprime sets

I am currently running into a problem related to coprime numbers. Consider a set of $d$-dimensional integer vectors, $z \subset \mathbb{Z}^d$ such that each component $z_i$ is bounded by another ...
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### Divide 500 into certain group so that all no's 1 to 500 can be found.

500 coins are there. Divide 500 coins into certain bags such that any rupees from 1 to 500 can be found by the combination of the bag's coins. What are the minimum nos of bags ?
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### Efficiently identifying spam honeypots

I realise that the title is computing specific, but I think the underlying problem is general - I just don't know how to phrase it more generally (which may be part of my problem). So I am asking ...
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### Divisibility of multinomial by a prime number

What is the condition for divisibility of multinomial $\dbinom {n}{x_1, x_2, \dots, x_k}$ by a prime $p$? Update: I tried to solve using a generalisation of Lucas Theorem by representing the $n$ ...
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### A question on primes and equal products

Is the following statement true; "For any odd prime number $p$ , any set of $p-1$ consecutive integers can not be partitioned into two subsets such that the elements of the two sets have equal ...
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### Proving that a “prime graph” is connected

Let the prime graph be defined as the graph of all natural numbers, with two vertices being connected if the sum of the numbers on the two vertices add up to a prime number. Prove that the prime ...
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### Counting the number of distinct integers in a range that fit a specified pattern

I've been thinking about primorials in the context of the twin prime conjecture. I am seeing this primarily as an exercise to improve my intuition about primorials and prime patterns more than the ...
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### Divisibility and factors [duplicate]

1) Can factors be negative? Please prove your opinion. 2)If prime factorization is given to you, how will you find out how many composite factors are there? Not the factors, just how many. For 2), my ...
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### Number of combinations

You are given K prime numbers, bigger than 6, find the number of different number that can be made of those prime numbers(using 1 number, 2 numbers ..., k numbers). Obviously you need to get the ...
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### How many ordered triple $(p,a,b)$ is possible such that $p^a=b^4+4$?

If we have a prime number $p$ and two natural numbers $a$ and $b$ such that $p^a=b^4+4$, then how many such ordered triplets $(p,a,b)$ exist? What should be the strategy to solve this one? The only I ...
How do I approach this problem using unique factorization?... How many numbers are product of (exactly) $3$ distinct primes $< 100$? edit: Just to add to that, How does unique factorization ...