Solution:** A number divisible by both 7 and 4 must be divisible by \( \text{lcm}(7, 4) = 28 \).
["Understanding Why Numbers Divisible by 7 and 4 Are Divisible by 28", "Mathematics often reveals elegant relationships between numbers, and one of the most interesting properties involves divisibility. A compelling example is the rule that a number divisible by both 7 and 4 must also be divisible by 28. Why is this the case? Let’s explore the underlying logic through the least common multiple (LCM).", "### What Does Divisible by 7 and 4 Mean?", "If a number is divisible by 7, it means it can be expressed as:\n[ N = 7k ]\nfor some integer ( k ).\nSimilarly, divisibility by 4 means:\n[ N = 4m ]\nfor some integer ( m ).", "For ( N ) to be divisible by both 7 and 4 simultaneously, it must satisfy both conditions at once — in other words, ( N ) must be a common multiple of 7 and 4.", "### The Least Common Multiple (LCM)", "The least common multiple (LCM) of two numbers is the smallest positive integer that is divisible by both. Instead of listing multiples, we calculate the LCM using prime factorization or the relationship between LCM and the greatest common divisor (GCD):\n[\n\ ext{lcm}(a, b) = \frac{a \ imes b}{\ ext{gcd}(a, b)}\n]", "For 7 and 4:\n- Both numbers are prime relative to each other (7 is prime, 4 = ( 2^2 )),\n- Their GCD is 1,", "So:\n[\n\ ext{lcm}(7, 4) = \frac{7 \ imes 4}{1} = 28\n]", "### Why Divisibility by 28 Follows", "Since 28 is the smallest number divisible by both 7 and 4, any number divisible by both must be a multiple of 28. This follows from the definition of LCM: any common multiple of 7 and 4 must be divisible by 28.", "For example:\n- ( 28 ) is divisible by 7 (( 28 \div 7 = 4 )) and by 4 (( 28 \div 4 = 7 )), so it satisfies the condition.\n- Any other multiple of 28, such as 56, 84, etc., also meets both divisibility requirements.", "But conversely, suppose a number is divisible by both 7 and 4 — it must be divisible by 28. This is because 28 captures the combined divisibility necessity without requiring extra factors.", "### Real-World Application", "This rule simplifies checking divisibility in number theory, cryptography, and algorithm design. Knowing that divisibility by both 7 and 4 implies divisibility by 28 can help streamline calculations and verify results efficiently.", "### Summary", "- A number divisible by 7 is divisible by 7.\n- A number divisible by 4 is divisible by 4.\n- Being divisible by both means the number is a common multiple.\n- The least common multiple of 7 and 4 is 28.\n- Hence, any number divisible by both 7 and 4 is divisible by 28.", "Understanding this relationship strengthens foundational number sense and highlights how LCM unlocks deeper mathematical insights.", "---", "Key takeaway:\nEvery number divisible by both 7 and 4 is divisible by ( \ ext{lcm}(7, 4) = 28 ), making 28 the smallest such candidate. This elegant property underpins important concepts in divisibility and modular arithmetic."]
