Find A & B: GCD And Sum Problems Solved!

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Find a & b: GCD and Sum Problems Solved!

Hey guys! Let's dive into some awesome math problems where we need to find two natural numbers, a and b, given their greatest common divisor (GCD) and their sum. We'll make sure a is less than b in all cases. Get ready, because we're about to break these problems down step by step!

Problem a) (a, b) = 34 and a + b = 272

Alright, let's kick things off with the first problem. We know that the greatest common divisor of a and b is 34, and their sum is 272. Here’s how we can solve it:

Since (a, b) = 34, we can express a and b as multiples of 34. Let a = 34x and b = 34y, where x and y are coprime (meaning their GCD is 1) and x < y. This is crucial because it ensures that a < b.

Now, we know that a + b = 272. Substituting our expressions for a and b, we get:

34x + 34y = 272

We can factor out 34 from the left side:

34(x + y) = 272

Now, divide both sides by 34:

x + y = 272 / 34

x + y = 8

So, we need to find two coprime numbers x and y that add up to 8, and remember, x < y. Let’s list the possible pairs of numbers that add up to 8:

(1, 7), (2, 6), (3, 5), (4, 4)

Now, we need to check which of these pairs are coprime:

  • (1, 7): GCD(1, 7) = 1 – This pair works!
  • (2, 6): GCD(2, 6) = 2 – This pair doesn’t work.
  • (3, 5): GCD(3, 5) = 1 – This pair works!
  • (4, 4): GCD(4, 4) = 4 – This pair doesn’t work, and also x must be less than y.

So, we have two valid pairs: (1, 7) and (3, 5).

For (1, 7):

  • a = 34 * 1 = 34
  • b = 34 * 7 = 238

For (3, 5):

  • a = 34 * 3 = 102
  • b = 34 * 5 = 170

Therefore, the two pairs of numbers that satisfy the conditions are (34, 238) and (102, 170).

Problem b) (a, b) = 15 and a + b = 1350

Next up, we have (a, b) = 15 and a + b = 1350. Let's tackle this one with a similar approach.

Since (a, b) = 15, we can express a and b as multiples of 15. Let a = 15x and b = 15y, where x and y are coprime and x < y.

We know that a + b = 1350. Substituting, we get:

15x + 15y = 1350

Factor out 15:

15(x + y) = 1350

Divide both sides by 15:

x + y = 1350 / 15

x + y = 90

Now, we need to find two coprime numbers x and y that add up to 90, with x < y. This might seem daunting, but let's think about it systematically.

We're looking for pairs (x, y) such that GCD(x, y) = 1. We can start by listing some possible values for x and then finding the corresponding y value:

  • If x = 1, then y = 89. GCD(1, 89) = 1. This works!
  • If x = 7, then y = 83. GCD(7, 83) = 1. This works!
  • If x = 11, then y = 79. GCD(11, 79) = 1. This works!
  • If x = 13, then y = 77. GCD(13, 77) = 1. This works!
  • If x = 17, then y = 73. GCD(17, 73) = 1. This works!
  • If x = 19, then y = 71. GCD(19, 71) = 1. This works!
  • If x = 23, then y = 67. GCD(23, 67) = 1. This works!
  • If x = 29, then y = 61. GCD(29, 61) = 1. This works!
  • If x = 31, then y = 59. GCD(31, 59) = 1. This works!
  • If x = 37, then y = 53. GCD(37, 53) = 1. This works!
  • If x = 41, then y = 49. GCD(41, 49) = 1. This works!
  • If x = 43, then y = 47. GCD(43, 47) = 1. This works!

So, here are the pairs: (1, 89), (7, 83), (11, 79), (13, 77), (17, 73), (19, 71), (23, 67), (29, 61), (31, 59), (37, 53), (41, 49), (43, 47) Let's calculate the pairs for (a, b): (15, 1335), (105, 1245), (165, 1185), (195, 1155), (255, 1095), (285, 1065), (345, 1005), (435, 915), (465, 885), (555, 795), (615, 735), (645, 705)

Problem c) (a, b) = 28 and a + b = 840

On to the next one! We have (a, b) = 28 and a + b = 840. Let's keep the ball rolling.

Since (a, b) = 28, let a = 28x and b = 28y, where x and y are coprime and x < y.

We know that a + b = 840. Substituting, we get:

28x + 28y = 840

Factor out 28:

28(x + y) = 840

Divide both sides by 28:

x + y = 840 / 28

x + y = 30

We need to find coprime numbers x and y that add up to 30, where x < y.

Let's list some possible values for x and find the corresponding y values:

  • If x = 1, then y = 29. GCD(1, 29) = 1. This works!
  • If x = 7, then y = 23. GCD(7, 23) = 1. This works!
  • If x = 11, then y = 19. GCD(11, 19) = 1. This works!
  • If x = 13, then y = 17. GCD(13, 17) = 1. This works!

So, here are the pairs: (1, 29), (7, 23), (11, 19), (13, 17) Let's calculate the pairs for (a, b): (28, 812), (196, 644), (308, 532), (364, 476)

Problem d) (a, b) = 7 and a + b = 735

Last but not least, we have (a, b) = 7 and a + b = 735. Let’s wrap this up!

Since (a, b) = 7, let a = 7x and b = 7y, where x and y are coprime and x < y.

We know that a + b = 735. Substituting, we get:

7x + 7y = 735

Factor out 7:

7(x + y) = 735

Divide both sides by 7:

x + y = 735 / 7

x + y = 105

We need to find coprime numbers x and y that add up to 105, where x < y.

Let's list some possible values for x and find the corresponding y values:

  • If x = 1, then y = 104. GCD(1, 104) = 1. This works!
  • If x = 2, then y = 103. GCD(2, 103) = 1. This works!
  • If x = 4, then y = 101. GCD(4, 101) = 1. This works!
  • If x = 8, then y = 97. GCD(8, 97) = 1. This works!
  • If x = 11, then y = 94. GCD(11, 94) = 1. This works!
  • If x = 13, then y = 92. GCD(13, 92) = 1. This works!
  • If x = 16, then y = 89. GCD(16, 89) = 1. This works!
  • If x = 17, then y = 88. GCD(17, 88) = 1. This works!
  • If x = 19, then y = 86. GCD(19, 86) = 1. This works!
  • If x = 22, then y = 83. GCD(22, 83) = 1. This works!
  • If x = 23, then y = 82. GCD(23, 82) = 1. This works!
  • If x = 26, then y = 79. GCD(26, 79) = 1. This works!
  • If x = 29, then y = 76. GCD(29, 76) = 1. This works!
  • If x = 31, then y = 74. GCD(31, 74) = 1. This works!
  • If x = 32, then y = 73. GCD(32, 73) = 1. This works!
  • If x = 34, then y = 71. GCD(34, 71) = 1. This works!
  • If x = 37, then y = 68. GCD(37, 68) = 1. This works!
  • If x = 38, then y = 67. GCD(38, 67) = 1. This works!
  • If x = 41, then y = 64. GCD(41, 64) = 1. This works!
  • If x = 43, then y = 62. GCD(43, 62) = 1. This works!
  • If x = 46, then y = 59. GCD(46, 59) = 1. This works!
  • If x = 47, then y = 58. GCD(47, 58) = 1. This works!
  • If x = 49, then y = 56. GCD(49, 56) = 1. This works!
  • If x = 52, then y = 53. GCD(52, 53) = 1. This works!

So, here are the pairs: (1, 104), (2, 103), (4, 101), (8, 97), (11, 94), (13, 92), (16, 89), (17, 88), (19, 86), (22, 83), (23, 82), (26, 79), (29, 76), (31, 74), (32, 73), (34, 71), (37, 68), (38, 67), (41, 64), (43, 62), (46, 59), (47, 58), (49, 56), (52, 53) Let's calculate the pairs for (a, b): (7, 728), (14, 721), (28, 707), (56, 679), (77, 658), (91, 644), (112, 623), (119, 616), (133, 602), (154, 581), (161, 574), (182, 553), (203, 532), (217, 518), (224, 511), (238, 497), (259, 476), (266, 469), (287, 448), (301, 434), (322, 413), (329, 406), (343, 392), (364, 371)

Conclusion

And there you have it! We've successfully found all the pairs of natural numbers (a, b) that satisfy the given conditions for each problem. Remember, the key is to express a and b in terms of their greatest common divisor and then use the sum to find the possible values. Keep practicing, and you'll become a math whiz in no time!