Solving Divisibility by 36 Questions

To determine if a number is divisible by 36, we can use the divisibility rules. A number is divisible by 36 if and only if it is divisible by two coprime factors of 36. The most common coprime factors used are 4 and 9, because their divisibility rules are simple.

So, we need to check each option to see if it is divisible by both 4 and 9.

Checking Divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

• Option 1: 47502. The last two digits form the number 02. $2 \div 4$ is not a whole number. So, 47502 is not divisible by 4.
• Option 2: 29412. The last two digits form the number 12. $12 \div 4 = 3$. So, 29412 is divisible by 4.
• Option 3: 54732. The last two digits form the number 32. $32 \div 4 = 8$. So, 54732 is divisible by 4.
• Option 4: 87064. The last two digits form the number 64. $64 \div 4 = 16$. So, 87064 is divisible by 4.

From the divisibility check by 4, we can eliminate option 1 (47502) as a possibility for being divisible by 36.

Checking Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9.

Now we check the remaining options (2, 3, and 4) for divisibility by 9.

• Option 2: 29412. Sum of digits = $2 + 9 + 4 + 1 + 2 = 18$. $18 \div 9 = 2$. So, 29412 is divisible by 9.
• Option 3: 54732. Sum of digits = $5 + 4 + 7 + 3 + 2 = 21$. $21 \div 9$ is not a whole number. So, 54732 is not divisible by 9.
• Option 4: 87064. Sum of digits = $8 + 7 + 0 + 6 + 4 = 25$. $25 \div 9$ is not a whole number. So, 87064 is not divisible by 9.

From the divisibility check by 9, we see that only option 2 (29412) is divisible by 9 among the numbers that were also divisible by 4.

Conclusion: Identifying the Number Divisible by 36

A number must be divisible by both 4 and 9 to be divisible by 36. Based on our checks:

• 47502 is not divisible by 4.
• 29412 is divisible by both 4 and 9.
• 54732 is divisible by 4 but not by 9.
• 87064 is divisible by 4 but not by 9.

Therefore, the only number divisible by 36 among the given options is 29412.

Revision Table: Key Divisibility Rules

Additional Information on Divisibility Tests

Divisibility rules are shortcuts to determine if a number can be divided by another number without performing the full division. The rule for checking divisibility by a composite number (like 36) often involves breaking it down into coprime factors.

• Coprime Factors: Two numbers are coprime if their greatest common divisor (GCD) is 1. For 36, pairs of coprime factors are (1, 36), (4, 9). Using (4, 9) is convenient due to their simple divisibility rules. Using non-coprime factors like (2, 18) wouldn't work because a number can be divisible by both 2 and 18 without being divisible by 36 (e.g., 18 is divisible by 2 and 18, but not 36).
• Applications: Divisibility tests are useful in simplifying fractions, finding factors, and solving problems in number theory and quantitative aptitude exams.
• Beyond Basic Rules: There are divisibility rules for other numbers too, though they can be more complex (e.g., for 7, 11, 13).