Find the value of x in 75 x 2 if it is divisible by 7.

The value of x is 5.

To find if 75x2 is divisible by 7, you need to calculate the product first and then check if it's divisible by 7.

  1. Calculate the product: First, calculate the product of 75 × 2. 75 × 2 = 150

  2. Check divisibility by 7: Now, we need to find a value of x such that 150 × x is divisible by 7.

  3. Trial and Error Method: We'll try different values of x starting from 1 and increasing until we find a value that results in a multiple of 7.

    • 150 × 1 = 150 (not divisible by 7)
    • 150 × 2 = 300 (not divisible by 7)
    • 150 × 3 = 450 (not divisible by 7)
    • 150 × 4 = 600 (not divisible by 7)
    • 150 × 5 = 750 (divisible by 7)

  4. Conclusion: We found that 150 × 5 = 750 is divisible by 7.

Therefore, the value of x in 75 × 2 such that the result is divisible by 7 is indeed x = 5.

Divisibility - Concept and Examples

Divisibility is a concept in number theory that describes the relationship between two integers. In simple terms, a number a is divisible by another number b if the division of a by b results in a whole number with no remainder.

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Here are some key points about divisibility:

  • Divisor: The number b by which we are dividing is called the divisor.
  • Dividend: The number a being divided is called the dividend.
  • Quotient: The whole number obtained from the division is called the quotient.
  • Remainder: In cases where the division is not exact, the remaining value after the division is called the remainder. When the remainder is zero, the dividend is divisible by the divisor.

For example, 12 is divisible by 3 because 12 divided by 3 equals 4, a whole number with no remainder.

Divisibility Rules

While performing actual division can be straightforward for smaller numbers, it can become cumbersome for larger ones. This is where divisibility rules come in handy. These rules are shortcuts that allow you to quickly determine if a number is divisible by another number, simply by examining its digits.

Here are some commonly used divisibility rules:

  • Divisibility by 2: A number is divisible by 2 if the last digit is even (0, 2, 4, 6, or 8).
  • Divisibility by 3: The sum of the digits of a number is divisible by 3 if the number itself is divisible by 3. For example, 15 is divisible by 3 because 1 + 5 = 6, which is divisible by 3.
  • Divisibility by 4: The last two digits of a number are divisible by 4 if the number itself is divisible by 4. For example, 1232 is divisible by 4 because 32 is divisible by 4.
  • Divisibility by 5: A number is divisible by 5 if the last digit is either 0 or 5.
  • Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3 (follows the rules for both divisibility by 2 and 3).
  • Divisibility by 9: Similar to divisibility by 3, the sum of the digits of a number is divisible by 9 if the number itself is divisible by 9.

These are just a few examples, and there are divisibility rules for other numbers as well. It's important to remember that these rules are not foolproof, and there may be exceptions in certain cases. However, they can be a valuable tool for quickly determining divisibility in many situations.