87 Divisibility Check

The question requires identifying which number among the given options—7221, 7099, 6712, and 7741—is divisible by 87. We will analyze each number based on divisibility rules.

Prime Factors of 87 Explained

To determine divisibility by 87, we first find its prime factors.

The number 87 can be factorized as: $87 = 3 \times 29$.

This means that for a number to be divisible by 87, it must be divisible by both 3 and 29.

Divisibility by 3 Test Results

A number is divisible by 3 if the sum of its digits is divisible by 3. Let's apply this rule to each option:

• 7221: The sum of the digits is $7 + 2 + 2 + 1 = 12$. Since $12 \div 3 = 4$, 7221 is divisible by 3.
• 7099: The sum of the digits is $7 + 0 + 9 + 9 = 25$. Since 25 is not divisible by 3, 7099 is not divisible by 3.
• 6712: The sum of the digits is $6 + 7 + 1 + 2 = 16$. Since 16 is not divisible by 3, 6712 is not divisible by 3.
• 7741: The sum of the digits is $7 + 7 + 4 + 1 = 19$. Since 19 is not divisible by 3, 7741 is not divisible by 3.

Based on the divisibility test for 3, only 7221 is divisible by 3. This implies that 7221 is the potential candidate for being divisible by 87.

Divisibility by 29 Test Results

Next, we check if 7221 is divisible by 29. We perform long division:

Check the division: $7221 \div 29$.

Step-by-step division:

1. Divide the first few digits of 7221 (72) by 29: $72 \div 29$. The largest multiple of 29 less than or equal to 72 is $2 \times 29 = 58$. The remainder is $72 - 58 = 14$.
2. Bring down the next digit (2) to make the number 142.
3. Divide 142 by 29: $142 \div 29$. The largest multiple of 29 less than or equal to 142 is $4 \times 29 = 116$. The remainder is $142 - 116 = 26$.
4. Bring down the next digit (1) to make the number 261.
5. Divide 261 by 29: $261 \div 29$. We find that $9 \times 29 = 261$. The remainder is $261 - 261 = 0$.

The division $7221 \div 29$ results in 249 with a remainder of 0.

$7221 \div 29 = 249$

Since there is no remainder, 7221 is divisible by 29.

7221 Divisibility by 87 Confirmation

As 7221 is divisible by both prime factors of 87 (3 and 29), it is confirmed to be divisible by 87.

We can verify this by performing the division of 7221 by 87:

$7221 \div 87$

Performing the division:

1. Divide 722 by 87: $722 \div 87$. The largest multiple of 87 less than or equal to 722 is $8 \times 87 = 696$. The remainder is $722 - 696 = 26$.
2. Bring down the next digit (1) to make 261.
3. Divide 261 by 87: $261 \div 87$. We find that $3 \times 87 = 261$. The remainder is $261 - 261 = 0$.

The division $7221 \div 87$ results in 83 with a remainder of 0.

$7221 \div 87 = 83$

Therefore, the number 7221 is divisible by 87.