Prove that 7∣3a+4b , 7∣5a−3b is 7∣13a−2b.

Sure, let's prove that if 7 divides 3a+4b and 7 divides 5a-3b, then 7 also divides 13a-2b.

Step 1: Given that 7 divides 3a+4b, we can write it as 3a+4b = 7k, where k is an integer.

Step 2: Similarly, given that 7 divides 5a-3b, we can write it as 5a-3b = 7l, where l is an integer.

Step 3: Now, we need to prove that 7 divides 13a-2b. Let's express 13a-2b in terms of 3a+4b and 5a-3b.

Step 4: We can write 13a-2b as 2*(3a+4b) + 3*(5a-3b).

Step 5: Substituting the values from step 1 and step 2, we get 2*(7k) + 3*(7l) = 14k + 21l = 7*(2k+3l).

Step 6: Since 2k+3l is an integer (as k and l are integers), we can say that 7 divides 13a-2b.

Therefore, we have proved that if 7 divides 3a+4b and 7 divides 5a-3b, then 7 also divides 13a-2b.