A1_n= 2+(n-1)*7=7n-5
A2_k= 3+(k-1)*9=9k-6
Put A1_n=A2_k => so 9k-7n=1 and solve the Diophantine equation for n,k u can see that gcd(9,7)=1 so the solutions you want come from applying the euclidean algorithm and your numbers n,k must be less than or equal to 1000.
How many integers appear in both of the following arithmetic progressions:
A1: 2, 9, 16,.....2+(1000-1)*7
A2: 3, 12, 21,.....3+(1000-1)*9
Since 2 appears in A1 but not in A2, it does not appear in both of the arithmetic progressions.
2 answers
Calculating GCD(9,7) gives:
7 = 0*9 + 7
9 = 1*7 + 2
7 = 3*2 + 1
2 = 2*1 + 0
Then applying the Extended Euclidean Algorithm:
1 = (1 * 7) + (-3 * 2)
= (-3 * 9) + (4 * 7)
= (4 * 7) + (-3 * 9)
A particular solution is:
x0 = -3
y0 = 4
The complete solution is:
x = -3 + 7n
y = 4 - 9n
7 = 0*9 + 7
9 = 1*7 + 2
7 = 3*2 + 1
2 = 2*1 + 0
Then applying the Extended Euclidean Algorithm:
1 = (1 * 7) + (-3 * 2)
= (-3 * 9) + (4 * 7)
= (4 * 7) + (-3 * 9)
A particular solution is:
x0 = -3
y0 = 4
The complete solution is:
x = -3 + 7n
y = 4 - 9n