Find all possible pairs of two natural numbers whose sum is 77 and their g.c.d. is 7
find all possible pairs of two natural numbers whose sum is 77and their g.c.d.is7.
a+b=77
n1*7+n2*7=77 where n1; n2 are natural numbers...
n1+n2=77/7=11
so pairs of n1,n2 can be
10,1
9,2
8,3 and so on
so the number pairs are
70,7
63,14
56,21 and so on....
a+b=77
n1+n2=77
77/7=11
Where n1,n2are natural numbers
So the pair of naturals are (7,70)(14,63)(21,56)(28,49) ...so on.
Find all possible pairs of two natural no whose sum is 77 and their g. C. D is 7
thank you for the answers
To find all possible pairs of two natural numbers whose sum is 77 and their greatest common divisor (GCD) is 7, we can follow these steps:
Step 1: List all the factors of 7.
The factors of 7 are 1 and 7.
Step 2: Identify the numbers whose sum is 77.
To find two numbers whose sum is 77, we can start with the first factor of 7, which is 1. We subtract 1 from 77 to get 76. So, we need to find two numbers whose sum is 76.
Step 3: Determine the pairs of numbers whose sum is 76.
We start by writing down the first pair: (1, 76). This means that 1 + 76 = 77.
Step 4: Check the GCD of each pair.
Now, we need to determine the GCD of each pair. If it is not equal to 7, we discard it. The GCD of any number with 1 is always 1. Therefore, we only need to check the GCD of the second number in the pair with 7. If it is equal to 7, then the pair satisfies the given condition.
Let's check each pair:
- (1, 76): GCD(76, 7) ≠ 7 (not satisfying)
- (2, 75): GCD(75, 7) = 1 (not satisfying)
- (3, 74): GCD(74, 7) ≠ 7 (not satisfying)
- (4, 73): GCD(73, 7) ≠ 7 (not satisfying)
- (5, 72): GCD(72, 7) ≠ 7 (not satisfying)
- (6, 71): GCD(71, 7) ≠ 7 (not satisfying)
- (7, 70): GCD(70, 7) = 7 (satisfying)
- (8, 69): GCD(69, 7) ≠ 7 (not satisfying)
- (9, 68): GCD(68, 7) ≠ 7 (not satisfying)
... and so on.
By checking each pair where the sum is 76 and calculating the GCD of the second number with 7, we can identify all possible pairs that satisfy the given conditions. In this case, the only satisfying pair is (7, 70).