87 Divisibility Check: Step-by-Step Solution

The question asks us to identify which number among the given options is divisible by 87. To solve this, we will use the concept of divisibility rules.

87 Divisibility Explained

The number 87 is a composite number. Its prime factorization is:

$87 = 3 \times 29$

Therefore, a number is divisible by 87 if and only if it is divisible by both 3 and 29.

3 Divisibility Test

First, let's apply the divisibility rule for 3 to each option. A number is divisible by 3 if the sum of its digits is divisible by 3.

• Option 1: 8004
  Sum of digits = $8 + 0 + 0 + 4 = 12$.
  Since $12 \div 3 = 4$, 12 is divisible by 3. Thus, 8004 is divisible by 3.
• Option 2: 8088
  Sum of digits = $8 + 0 + 8 + 8 = 24$.
  Since $24 \div 3 = 8$, 24 is divisible by 3. Thus, 8088 is divisible by 3.
• Option 3: 7150
  Sum of digits = $7 + 1 + 5 + 0 = 13$.
  Since 13 is not divisible by 3, 7150 is not divisible by 3.
• Option 4: 7835
  Sum of digits = $7 + 8 + 3 + 5 = 23$.
  Since 23 is not divisible by 3, 7835 is not divisible by 3.

From this test, we can eliminate options 3 (7150) and 4 (7835). We only need to check options 1 (8004) and 2 (8088) further.

29 Divisibility Test

Next, we need to check if the remaining numbers, 8004 and 8088, are divisible by 29. We can perform direct division.

• Checking 8004:
  Divide 8004 by 29.
  $8004 \div 29$
  The division yields 276 with no remainder:
  $8004 = 29 \times 276$
  So, 8004 is divisible by 29.
• Checking 8088:
  Divide 8088 by 29.
  $8088 \div 29$
  The division yields 278 with a remainder of 26:
  $8088 = 29 \times 278 + 26$
  So, 8088 is not divisible by 29.

87 Divisibility Confirmation

We summarize our findings:

• 8004: Divisible by 3? Yes. Divisible by 29? Yes. Therefore, divisible by 87? Yes.
• 8088: Divisible by 3? Yes. Divisible by 29? No. Therefore, divisible by 87? No.
• 7150: Divisible by 3? No. Therefore, divisible by 87? No.
• 7835: Divisible by 3? No. Therefore, divisible by 87? No.

The only number that satisfies both conditions (divisibility by 3 and 29) is 8004. Hence, 8004 is divisible by 87.