This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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7. Without dividing, show that 524 is divisible by 4. Step 1: Recall the divisibility rule for 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4. Step 2: Identify the last two digits of 524, which are 24. Step 3: Check if 24 is divisible by 4. 24 ÷ 4 = 6 Since 24 is divisible by 4, the number 524 is also divisible by 4. Therefore, 524 is divisible by 4. 8. List the composite numbers less than 12. Step 1: Recall that a composite number is a positive integer that has at least one divisor other than 1 and itself. Numbers 0 and 1 are neither prime nor composite. Step 2: List integers less than 12: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Step 3: Identify prime numbers (only divisible by 1 and themselves): 2, 3, 5, 7, 11. Step 4: The remaining numbers (excluding 1) are composite. The composite numbers less than 12 are 4, 6, 8, 9, 10. Number pattern and sequence. 1. Find the next number in the sequence. a) 2, 3, 5, 7, 9 Step 1: Analyze the differences between consecutive terms. 3 - 2 = 1 5 - 3 = 2 7 - 5 = 2 9 - 7 = 2 Step 2: The pattern is an initial increase of 1, followed by consistent increases of 2. Step 3: Apply the pattern to find the next term. The next number is 9 + 2 = 11. The next number in the sequence is 11. b) 0, 2, 4, 6, 8, 10 Step 1: Analyze the differences between consecutive terms. 2 - 0 = 2 4 - 2 = 2 6 - 4 = 2 8 - 6 = 2 10 - 8 = 2 Step 2: The pattern is a consistent increase of 2 (an arithmetic progression). Step 3: Apply the pattern to find the next term. The next number is 10 + 2 = 12. The next number in the sequence is 12. 2. The sum of three consecutive even numbers is 48. Find the numbers. Step 1: Let the first even number be x. Since the numbers are consecutive even numbers, the next two even numbers will be x+2 and x+4. Step 2: Set up an equation based on the given sum. x + (x+2) + (x+4) = 48 Step 3: Combine like terms and solve for x. 3x + 6 = 48 3x = 48 - 6 3x = 42 x = (42)/(3) x = 14 Step 4: Find the three consecutive even numbers. The first number is x = 14. The second number is x+2 = 14+2 = 16. The third number is x+4 = 14+4 = 18. The three consecutive even numbers are 14, 16, 18. 3. Which number when divided by 12 and 8 leaves 2 as a reminder? Step 1: Find the Least Common Multiple (LCM) of 12 and 8. Multiples of 12: 12, 24, 36, ... Multiples of 8: 8, 16, 24, 32, ... The LCM of 12 and 8 is 24. Step 2: The number that is exactly divisible by both 12 and 8 is their LCM, which is 24. Step 3: To find a number that leaves a remainder of 2 when divided by 12 and 8, add 2 to the LCM. 24 + 2 = 26 The number is 26. 4. The L.C.M of 2 numbers is 36 and their GCF is 6, if the first number is 12, find the second number. Step 1: Use the relationship between LCM, GCF, and two numbers (N_1, N_2): N_1 × N_2 = LCM(N_1, N_2) × GCF(N_1, N_2) Step 2: Substitute the given values into the formula. Let the first number (N_1) be 12. Let the second number (N_2) be x. LCM = 36 GCF = 6 12 × x = 36 × 6 Step 3: Solve for x. 12x = 216 x = (216)/(12) x = 18 The second number is 18. Last free one today — make it count tomorrow, or type /upgrade for unlimited.