What is the lowest square number divisible by 28?

The lowest square number (perfect square) divisible by 28 is 196, which equals 14². To understand why this is the answer and how to find it systematically, we need to examine the prime factorization of 28 and apply the rules for perfect squares. This type of problem tests your understanding of both divisibility and the properties of perfect squares, making it a valuable exercise in Class 10 number theory.

The systematic approach to finding this answer involves analyzing the prime factorization of 28. Breaking down 28: 28 = 4 × 7 = 2² × 7. For a number to be divisible by 28, it must contain all the prime factors of 28 with at least the same exponents—that is, it must have at least 2² and at least 7¹ in its factorization. Additionally, for this number to be a perfect square, all prime factors must appear with even exponents (forming complete pairs). Currently, we have 2² (even exponent—good for being a perfect square) and 7¹ (odd exponent—problematic for being a perfect square). To make this a perfect square while maintaining divisibility by 28, we need to add one more factor of 7 to create 7² (an even exponent). Therefore, the smallest perfect square divisible by 28 must have the factorization 2² × 7² = 4 × 49 = 196.

We can verify this answer: First, check that 196 is a perfect square: √196 = 14, which is a whole number, confirming 196 = 14². Second, check divisibility: 196 ÷ 28 = 7, which is a whole number, confirming that 28 divides evenly into 196. Third, verify it's the smallest: Any smaller perfect square wouldn't be divisible by 28. For instance, the next smaller perfect square is 169 (13²), and 169 ÷ 28 = 6.036..., which is not a whole number, so 169 is not divisible by 28. Therefore, 196 is indeed the lowest square number divisible by 28.

This problem illustrates an important general principle in number theory: to find the lowest perfect square divisible by a given number n, first find the prime factorization of n, then adjust all odd exponents to the next even number (by multiplying by the appropriate prime factors), and leave even exponents as they are. The resulting number is your answer. Let's apply this to another example: What's the lowest perfect square divisible by 72? Prime factorization: 72 = 2³ × 3². The exponent of 2 is 3 (odd), so we need one more factor of 2 to make it 2⁴ (even). The exponent of 3 is already 2 (even), so no change needed. Therefore, the lowest perfect square divisible by 72 is 2⁴ × 3² = 16 × 9 = 144 = 12². This systematic approach works for any number.