# Thread: Least number which leaves remainder 1,2,3,4 when divided succesively by 2,3,5,7

1. ## Least number which leaves remainder 1,2,3,4 when divided succesively by 2,3,5,7

Hi All,

What is the least number which leaves remainder 1, 2, 3 and 4 when divided successively by 2, 3, 5 and 7?

Thank you,
Ashwin Kumar D P.
Bangalore, India.

2. Originally Posted by ashwinkumardp
Hi All,
What is the least number which leaves remainder 1, 2, 3 and 4 when divided successively by 2, 3, 5 and 7?
Thank you,
Ashwin Kumar D P.
Bangalore, India.
$\displaystyle x \equiv 1 \mod 2$ Equation1

$\displaystyle x \equiv 2 \mod 3$ Equation2

$\displaystyle x \equiv 3 \mod 5$ Equation3

$\displaystyle x \equiv 4 \mod 7$ Equation4

Use the Chinese Remainder Theorem.

However, for this question try this:
The values that satisfy Eqn4 are
4, 11, 18, 25, 32, 39, ...
They are 4 + multiples of 7.

Use those values in Eqn3

$\displaystyle 4 \equiv 4 \mod 5$

$\displaystyle 11 \equiv 1 \mod 5$

$\displaystyle 18 \equiv 3 \mod 5$

18 satisfies Eqn4 & Eqn3

Use 18 + multiples of 35 ( 7 times 5 ) for Equation2
18, 53, 88, 123, ...

Find the one that satisfies equation2.
Then a similar procedure for Equation1

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