a = 36, b - 60 a = 901, b = 935 a = 72, b = 714 a = 12628, b = 21361 a = -36, b

Question

a = 36, b - 60 a = 901, b = 935 a = 72, b = 714 a = 12628, b = 21361 a = -36, b = -60 a = 901,b = -935 Find integers u and v such that 9u + 14u = 1 or explain why it is not possible to do so. Then find integers x and y such that 9x + 14y = 10 or explain why it is not possible to do so. Find integers x and y such that 9x + 15y = 10 or explain why it is not possible to do so. Find integers x and y such that 9x + 15y = 3162 or explain why it is not possible to do so. Notice that gcd(11, 17) = 1. Find integers x and y such that 11x + 17y = 1. Let m, n Z. Write the sum m / 11 + n / 17 as a single fraction. Find two rational numbers with denominators of 11 and 17. respectively . whose sum is equal to 10 / 187 Hint: Write the rational numbers in the form m / 11 and n / 17, where m , n Z. Then write

Explanation / Answer

a)

14 = 1*9 + 5

9 = 1*5 + 4

5 = 4*1 + 1

Therefore:

1 = 5-4 = 5 - (9 - 5) = 2*5 - 9 = 2*(14 - 9) - 9 = 2*14 - 3*9 -> u = -3 , v = 2

Multiply both sides by 10:

10 = 20*14 - 30*9 -> u = -30 , v = 20

b)

gcd(9,15) = 3 but 3 does not divide 10, so it is not possible.

c)

gcd(9,15) = 3 and 3|3162 -> possible

15 = 1*9 + 6

9 = 1*6 + 3

6 = 2*3 -> 3 = 9 - 6 = 9 - (15 - 9) = 2*9 - 15

3162 = 3*1054 -> multiply both sides by 1054:

3*1054 = (2*1054)*9 - 1*1054 * 15

3162 = 2108*9 - 1054*15

-> x = 2108 , y = -1054

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