Question

A number when divided successively by 4 5 and 6 and remainder is 2,3,4 what is the number

Answer 1

____________________________________

∆ Let the number be 'X'

• Recall : Euclid's theorem... :
$a = bq + r \: \: \: \: \: \: \\ \\ 0 \leqslant r < b$

✓✓ X = 4a + 2
✓✓ X = 5b + 3
✓✓ X = 6c + 4

Adding two in each equation :

✓✓ ( X + 2 ) = 4( a + 1 )
✓✓ ( X + 2 ) = 5( b + 1 )
✓✓ ( X + 2 ) = 6( c + 1 )

=> ( X + 2 ) is the LCM of 4, 5 and 6
=> ( X + 2 ) = 60
=> ( X ) = 58
____________________________________

However, when you look at it, the number 58 might not be the only number that satisfies this. Let's look at LCM of 4, 5 and 6 again.

• LCM = 60
∆ 4, 5 and 6 all the three divides this

=> The Period of occurrence of the solution is '60'

=> The answer is ( 60k + 58 )
____________________________________

∆ The number which satisfies the above condition is : ( 60k + 58 ) where 'k' is any whole number