7109
7
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Answers
Answer:
7,109 (seven thousand one hundred nine) is an odd four-digits prime number following 7108 and preceding 7110. In scientific notation, it is written as 7.109 × 103. The sum of its digits is 17. It has a total of 1 prime factor and 2 positive divisors. There are 7,108 positive integers (up to 7109) that are relatively prime to 7109.
Answer:
In most of the programming competitions, we are required to answer the result in 10^9+7 modulo. The reason behind this is, if problem constraints are large integers, only efficient algorithms can solve them in allowed limited time.
What is modulo operation:
The remainder obtained after the division operation on two operands is known as modulo operation. Operator for doing modulus operation is ‘%’. For ex: a % b = c which means, when a is divided by b it gives the remainder c, 7%2 = 1, 17%3 = 2.
Why do we need modulo:
The reason of taking Mod is to prevent integer overflows. The largest integer data type in C/C++ is unsigned long long int which is of 64 bit and can handle integer from 0 to (2^64 – 1). But in some problems where the growth rate of output is very high, this high range of unsigned long long may be insufficient.
Suppose in a 64 bit variable ‘A’, 2^62 is stored and in another 64 bit variable ‘B’, 2^63 is stored. When we multiply A and B, the system does not give a runtime error or exception. It just does some bogus computation and stores the bogus result because the bit size of result comes after multiplication overflows.
In some of the problems, to compute the result modulo inverse is needed and this number helps a lot because it is prime. Also this number should be large enough otherwise modular inverse techniques may fail in some situations.
Due to these reasons, problem setters require to give the answer as a result of modulo of some number N.
There are certain criteria on which value of N depends:
It should just be large enough to fit in the largest integer data type i.e it makes sure that there is no overflow in result.
It should be a prime number because if we take mod of a number by Prime the result is generally spaced i.e. the results are very different results in comparison to mod the number by non-prime, that is why primes are generally used for mod.
10^9+7 fulfills both the criteria. It is the first 10-digit prime number and fits in int data type as well. In fact, any prime number less than 2^30 will be fine in order to prevent possible overflows.
How modulo is used:
A few distributive properties of modulo are as follows:
( a + b) % c = ( ( a % c ) + ( b % c ) ) % c
( a * b) % c = ( ( a % c ) * ( b % c ) ) % c
( a – b) % c = ( ( a % c ) – ( b % c ) ) % c
( a / b ) % c = ( ( a % c ) / ( b % c ) ) % c
So, modulo is distributive over +, * and – but not over / [Please refer Modular Division for details]
NOTE: The result of ( a % b ) will always be less than b.
In the case of computer programs, due to the size of variable limitations, we perform modulo M at each intermediate stage so that range overflow never occurs.
Example:
a = 145785635595363569532135132
b = 3151635135413512165131321321
c = 999874455222222200651351351
m = 1000000007
Print (a*b*c)%m.
Method 1:
First, multiply all the number and then take modulo:
(a*b*c)%m = (459405448184212290893339835148809
515332440033400818566717735644307024625348601572) %
1000000007
a*b*c does not fit even in the unsigned long long
int due to which system drop some of its most
significant digits. Therefore, it gives the wrong answer.
(a*b*c)%m = 798848767
Method 2:
Take modulo at each intermediate steps:
i = 1
i = (i*a) % m // i = 508086243
i = (i*b) % m // i = 144702857
i = (i*c) % m // i = 798848767
i = 798848767
Method 2 always gives the correct answer.
Step-by-step explanation:
I Hope It Helps