Remainder in AP?

Number Theory Level pending

Rahul decides to open his father's laptop. But it is password protected. His father being a mathematician has left a mathematical hint for the password. It had the following contents:

Password is a natural number and it is the least number satisfying the conditions given below.

  • The number when divided by 2 leaves remainder 0
  • The number when divided by 3 leaves remainder 1
  • The number when divided by 4 leaves remainder 2
  • The number when divided by 5 leaves remainder 3
  • The number when divided by 6 leaves remainder 4
  • The number when divided by 7 leaves remainder 5
  • The number when divided by 8 leaves remainder 6
  • The number when divided by 9 leaves remainder 7
  • The number when divided by 10 leaves remainder 8

Find the password.


The answer is 2518.

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Akshat Joshi by Akshat Joshi , 20, India

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1 solution

Maria Kozlowska
Feb 26, 2017

We need the smallest natural number divisible by integers 2 2 to 10 10 , increased by 2 2 . The number is 2 3 × 3 2 × 5 × 7 + 2 = 2518 2^3 \times 3^2 \times 5 \times 7 +2=\boxed{2518} .

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