Solution:
To solve the problem, we will use the Chinese remainder theorem.

Step 1: Find the factors of 841
We can find the factors of 841 by prime factorization. 841 = 29 x 29.

Step 2: Express the remainder in terms of the factors
Let the number be x. We know that x leaves a remainder of 87 when divided by 841. This can be expressed as:

x ≡ 87 (mod 841)

Step 3: Apply the Chinese remainder theorem
We need to find the remainder of x when divided by 29. To do this, we will use the Chinese remainder theorem.

Since 841 = 29 x 29, we can express x as:

x ≡ a (mod 29)
x ≡ b (mod 29)

where a and b are the remainders when x is divided by 29.

We can find a and b by using the fact that x ≡ 87 (mod 841). We know that x is 87 more than a multiple of 841. So we can express x as:

x = 841n + 87

where n is an integer.

Substituting this in the first congruence, we get:

841n + 87 ≡ a (mod 29)

Multiplying both sides by 29, we get:

24389n + 2523 ≡ a (mod 29)

Since 24389 is a multiple of 29, we can simplify this to:

2523 ≡ a (mod 29)

Similarly, substituting x in the second congruence, we get:

841n + 87 ≡ b (mod 29)

Multiplying both sides by 29, we get:

24389n + 2523 ≡ b (mod 29)

Since 24389 is a multiple of 29, we can simplify this to:

2523 ≡ b (mod 29)

So we have:

x ≡ 2523 (mod 29)

Step 4: Find the remainder
We have found that x ≡ 2523 (mod 29). So the remainder when x is divided by 29 is 2523.

But we need to check if this is correct. We know that x ≡ 87 (mod 841). So we need to check if x ≡ 2523 (mod 29) and x ≡ 87 (mod 841) are both satisfied.

We can check this by expressing x as:

x = 841n + 87

Substituting x = 841n + 87 in the second congruence, we get:

841n + 87 ≡ 87 (mod 841)

This is true for any value of n.

Substituting x = 2523 + 29m in the first congruence, we get:

841n + 87 ≡ 2523 + 29m (mod 841)

Multiplying both sides by 29, we get:

24389n + 2523 ≡ 2523 + 29m (mod 841)

Simplifying, we get:

24389n ≡ 29m (mod 841)

Dividing both sides by 29, we get:

841n ≡ m (mod 29)

Since 841 is a multiple of 29, we can