If m>1 divides (7n+3)
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19 May 2022, 05:42
Concept:
1. If m divides n, then n is a divisor or factor of m.
Example: If 2 divides 4, then we can say that 2 is a factor of 4. (4 = 2 x 2)
2. If m is a factor of n, then m is also a factor of (K) x n Here K = {0,1,2,3....}
If m is a factor of n, then m is also a factor of (2) x n (Assumed K = 2 for example)
Example: If 3 is a factor of 9, then 3 is also a factor of (2) x 9, (3) x 9, (4) * 9 and so on.
3. If m is a factor of n and m is a factor of p, then m is a factor of m + p and m - p.
Example: If 3 is a factor of 9, 3 is a factor of 12, then 3 is also a factor of (3 + 12 = 15) and (3 - 12 = 9)
Coming to the question
m divides 7n + 3 implies that m is a factor of 7n + 3
m divides 35n + 26 implies that m is a factor of 35n + 26
As per the concept - 2,
If m is a factor of 7n + 3, then m is also a factor of (5)(7n + 3) - Agree ?
Which implies m is a factor of 35n + 15
As per the concept - 3,
If m is a factor of 35n + 26 and from the above equation
m is also a factor of 35n + 15
we can say that m is a factor 35n + 26 - (35n + 15) implies m is a factor of 11.
Now we know that m is a factor of 11, m = {11, 22, 33, 44,...}
That means we can say that 7n + 3 and 35n + 26 must be divisible by 11.