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
Details and assumptions
Since 2 appears in A1 but not in A2, it does not appear in both of the arithmetic progressions.

To find out how many integers appear in both of the given arithmetic progressions, we need to first find the number of terms in each progression and then identify the common integers.

1. Finding the number of terms in each arithmetic progression:
We are given the arithmetic progressions A1 and A2 with the formulas:
A1: 2, 9, 16, ......, 2 + (1000 - 1) × 7
A2: 3, 12, ......, 21, ......, 3 + (1000 - 1) × 9

The formula to find the nth term of an arithmetic progression is: an = a1 + (n - 1) × d
where an represents the nth term, a1 is the first term, n is the number of terms, and d is the common difference.

Let's find the number of terms in each progression:

For A1:
a1 = 2 (given)
d = 7 (common difference)
an = 2 + (1000 - 1) × 7 = 2 + 6993 = 6995 (last term)
We can use the formula to find the number of terms:
n1 = (an - a1 + d) / d
n1 = (6995 - 2 + 7) / 7 = 6999 / 7 = 999

For A2:
a1 = 3 (given)
d = 9 (common difference)
an = 3 + (1000 - 1) × 9 = 3 + 8991 = 8994 (last term)
Using the formula for the number of terms:
n2 = (an - a1 + d) / d
n2 = (8994 - 3 + 9) / 9 = 8991 / 9 = 999

2. Identifying the common integers:
Now that we know both progressions have 999 terms, we can compare the integers in each progression to find the common ones.

For A1, we have the terms: 2, 9, 16, ......, 6995
For A2, we have the terms: 3, 12, ......, 21, ......, 8994

To find the common integers, we need to consider the multiples of 7 in A1 and the multiples of 9 in A2. The common terms will be the ones that are both multiples of 7 and multiples of 9.

Let's find the largest multiple of 7 and 9 that falls within the range of 2 to 6995:
The largest multiple of 7 within this range is 6993 (999 × 7). However, it is not a multiple of 9, so it is not a common integer.

Now, let's find the largest multiple of 9 within the same range:
The largest multiple of 9 is 6993 (777 × 9), which falls within the range and is a multiple of both 7 and 9.

Therefore, the common integers appearing in both arithmetic progressions A1 and A2 are:
6993

So, only one integer, 6993, appears in both of the given arithmetic progressions.