SOLUTION: Which number who divided by 7,9,11and remainder 1,2,3?

Question 1052975: Which number who divided by 7,9,11and remainder 1,2,3?
Found 3 solutions by jim_thompson5910, ikleyn, advanced_Learner:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Let x be this unknown number that satisfies all three conditions given. For the sake of simplicity, we'll assume x is positive.

Use the Chinese Remainder Theorem

I will use the same basic steps and notation as that PDF is showing.

We have these 3 congruences given
x = 1 (mod 7)
x = 2 (mod 9)
x = 3 (mod 11)

The values are the right hand side values (1, 2 and 3 respectively) where 'i' refers to the modulus value

Multiply out the modulus values
N = 7*9*11 = 693

Calculate the values

Then finally we need to compute the multiplicative inverses values of each value

(mod 7)
Note: 99 = 1 (mod 7) so the solution to 1y = 1 (mod 7) is y = 1

(mod 9)
Note: 77 = 5 (mod 9). The solution to 5y = 1 (mod 9) is y = 2

(mod 11)
Note: 63 = 8 (mod 11) and the solution to 8y = 1 (mod 11) is y = 7

Now we use the information computed above to get the solution for x

(mod 693)

(mod 693)

(mod 693)

(mod 693)

The smallest positive solution for x is 344

The set of all solutions for x is the set of x values such that where and is an integer. In other words, x = 344 (mod 693)

Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.
Which number when divided by 7,9,11 gives the remainders 1,2,3 ?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The numbers that give the remainder 1 when divided by 7, form an arithmetic progression 1, 8, 15, 22, 29, . . . (1 + 7(n-1)), . . . Of these numbers, the smallest number that gives the remainder 2 when divided by 9, is . Next, the numbers that give the remainder 1 when divided by 7 and the remainder 2 when divided by 9, form an arithmetic progression 29, 29+63 = 92, 29+2*63 = 155, 29+3*63 = 218, . . . (29+(n-1)*63), . . . (notice: 63 = 7*9) Among them, the smallest number that gives the remainder 3 when divided by 11, is . Answer. Your number is 344.

Answer by advanced_Learner(501)   (Show Source): You can put this solution on YOUR website!
Which number who divided by 7,9,11and remainder 1,2,3?
............................................................................
let the number be x then
remainder is
remainder is
remainder is
remainder is
remainder is
remainder is
remainder is
remainder is
remainder is
meaning 2x+5 is divisble by
=
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