Modulo makes it weak

Number Theory Level 3

Consider a smallest positive integer P P which is divisible by all natural numbers from 1 1 to 100 100 inclusive. If b b is a non-negative integer, P P satisfy the congruence P ≡ b ( m o d 9699690 ) P \equiv b \pmod {9699690} Find the smallest value of b b .

Details and Assumptions

  • You may need to refer to list of primes as a reference.

  • No computational aid is required in solving this problem.


The answer is 0.

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Nihar Mahajan by Nihar Mahajan , 20, India

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2 solutions

Nihar Mahajan
Mar 26, 2015

Notice that 9699690 is the product of all prime numbers from 1 to 20. \text{Notice that 9699690 is the product of all prime numbers from 1 to 20.}

Also , P is divisible by all prime numbers from 1 to 100. \text{Also , P is divisible by all prime numbers from 1 to 100.}

Since all prime numbers are co-prime to each other , \text{Since all prime numbers are co-prime to each other ,}

9699690 ∣ P 9699690 | P

⇒ b = 0 \Rightarrow b = 0

9 Helpful 0 Interesting 0 Brilliant 0 Confused

THIS QUESTION Made my day. Perfecto!

Mehul Arora - 6 years, 2 months ago

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Thanks a lot!!!!

Nihar Mahajan - 6 years, 2 months ago

AWESOME Question and AWESOME Solution

LOVED IT !!!!!!!!

Vaibhav Prasad - 6 years, 2 months ago

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Thanks a lot!!!!

Nihar Mahajan - 6 years, 2 months ago
Sharky Kesa
May 2, 2015

I used a simple Python code to solve the problem. 

from fractions import gcd
print reduce(lambda a,b: a*b/gcd(a,b), range(1,100,1))%9699690
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