Question 3 of 29intermediate🔧 ApplyNumerical3 marks

Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.

Correct Answer

Given: Find the smallest square number divisible by 4, 9, and 10.

Step 1: Find the LCM of 4, 9, and 10 LCM(4, 9, 10) = 180

Step 2: Prime factorization of 180 180 = 2² × 3² × 5

Step 3: For a perfect square, all prime factors must have even exponents The factor 5 has exponent 1

Step 4: Multiply by 5 to make it a perfect square 180×5180 \times 5 = 900 = 2² × 3² × 5² = 30²

Answer: The smallest square number divisible by 4, 9, and 10 is 900.

Exercise: Figure it Out (Section 1) | Q: 5 | (Page Page 10)
For More Understanding

Explanation

The chapter explains that a number is a perfect square if its prime factors can be split into two identical groups, meaning all prime factors must have even exponents. To find the smallest square divisible by given numbers, we first find their LCM, then multiply by necessary factors to make all exponents even.

Solution Steps

  1. Step 1: Find LCM of 4, 9, 10

  2. Step 2: 4 = 2², 9 = 3², 10 = 2×52 \times 5

  3. Step 3: LCM = 2² × 3² × 5 = 180

  4. Step 4: For a perfect square, all exponents must be even

  5. Step 5: 180 = 2² × 3² × 5¹, need one more 5

  6. Step 6: Smallest square = 180×5180 \times 5 = 900 = 30²

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