If positive number x, y and z are divided by 34. The remainders are 19, 21 and 29 respectively when (7x−3y + 5z) is divided by 17?

Question

This is a question from the chapter number system.

Answer 1

Here, x=53 , y=55 and z=63

According to the question, the updated eqn will be:

7(53)-3(55)+5(63) = 521

We can write it as, 521 = (17 x 30) + 11.

Hence,the remainder is 11.

Answer 2

The remainder 19, 21 & 29 are x, y & z respectively.

      x = 19 , y= 21, z=29

    = (  7×19 - 3×21 + 5×29 ) ÷ 17

    = (153  - 63  + 145) ÷ 17

    = 235 ÷ 17

    = 13. 23

Answer 3

Given:

x ≡ 19 (mod 34)

y ≡ 21 (mod 34)

z ≡ 29 (mod 34)

We want to find the remainder when (7x - 3y + 5z) is divided by 17.

Step 1: Substitute the congruences into (7x - 3y + 5z):

(7x - 3y + 5z) ≡ 7(19) - 3(21) + 5(29) (mod 17)

Step 2: Calculate the expression:

(7x - 3y + 5z) ≡ 133 - 63 + 145 (mod 17)

Step 3: Simplify further:

(7x - 3y + 5z) ≡ 215 (mod 17)

The remainder when (7x - 3y + 5z) is divided by 17 is 215.

Answer 4

The given information relates to the Chinese Remainder Theorem. It states that if you have a system of linear congruences, such as:

css

x ≡ a (mod m)

y ≡ b (mod n)

z ≡ c (mod p)

Then the solution for (ax + by + cz) ≡ d (mod lcm(m, n, p)).

In this case:

lua

7x - 3y + 5z ≡ d (mod 17)

Given remainders:

lua

x ≡ 19 (mod 34)

y ≡ 21 (mod 34)

z ≡ 29 (mod 34)

You can substitute these into the equation and solve for d using the Chinese Remainder Theorem formula.

For precise numerical values, calculations are required.

Answer 5

If a number x is divided by a and it leaves remainder b, we can write x=a*n+b(here n is any integer)

taking n as 1,

Let the numbers be

x=34*1+19=53

y=34*1+21=55

z=34*1+29=63

thus,

7x-3y+5z=7(53)-3(55)+5(63)=521

521=17(30)+11

thus remainder is 11.

Answer 6

We are given that when x is divided by 34, the remainder is 19. This can be written as x ≡ 19 (mod 34).

Similarly, when y is divided by 34, the remainder is 21, which can be written as y ≡ 21 (mod 34).

Lastly, when z is divided by 34, the remainder is 29, which can be written as z ≡ 29 (mod 34).

We need to find the remainder when (7x - 3y + 5z) is divided by 17.

We can substitute the given congruences for x, y, and z into the expression:

(7x - 3y + 5z) ≡ (7(19) - 3(21) + 5(29)) (mod 17)

Simplifying this expression gives:

(7x - 3y + 5z) ≡ (133 - 63 + 145) (mod 17)

(7x - 3y + 5z) ≡ 215 (mod 17)

To find the remainder when 215 is divided by 17, we divide 215 by 17 and take the remainder:

215 ÷ 17 = 12 remainder 11

Therefore, the remainder when (7x - 3y + 5z) is divided by 17 is 11.

Answer 7

Here we have

X = (k × 34) + 19        ( equation 1) 

Y = (p × 34) +21        (equation 2)

Z= (q×34) +29           (equation  3) 

Now, from (equation 1)     7X = (7k × 34) + 133

          from (equation 2)     3Y = (3p × 34) + 81

          from (equation  3)    5Z = ( 5q × 34) + 145

Thus 7X - 3Y +5Z =  (7k × 34) + 133 - ((3p × 34) + 81) +( 5q × 34) + 145 = [(7k × -3p × 5q) × 34 ] + [133-81+145]

                  =  [(-105kpq) × 34 ] + 197

 Then, [(-105kpq) × 34 ] + 197 = [(-105kpq) × (2×17) ] + [(11×17) +10] = [ (-210kpq ×17) +[(11×17) +10]

                         =  (-210kpq +11 )×17 +10

Therefore the required remainder becomes 10.

 

   

Answer 8

We know that x leaves a remainder of 19 when divided by 34, y leaves a remainder of 21, and z leaves a remainder of 29.

Let's start by simplifying the expression (7x - 3y + 5z) by substituting in the remainders we have:

(7x - 3y + 5z) = (7(34a + 19) - 3(34b + 21) + 5(34c + 29))

where a, b, and c are integers representing the number of times each respective number has been divided by 34.

Simplifying this expression, we get:

(7x - 3y + 5z) = (238a + 115 - 102b - 63 + 170c + 145)

Combining like terms, we get:

(7x - 3y + 5z) = (238a - 102b + 170c + 197)

Now we can take this expression and divide it by 17 to find the remainder:

(7x - 3y + 5z) ÷ 17 = (238a - 102b + 170c + 197) ÷ 17

The remainder on the left-hand side will be the same as the remainder on the right-hand side.

We can simplify the right-hand side by dividing each term by 17:

(238a - 102b + 170c + 197) ÷ 17 = (14a - 6b + 10c + 11)

Therefore, the remainder when (7x - 3y + 5z) is divided by 17 is 11.