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.
ScanSolve AI Answer
To find which of the given numbers, when divided by 42, yields no fraction, we need to identify the number that is perfectly divisible by 42 (i.e., leaves a remainder of 0). A number is divisible by 42 if it is divisible by both 6 and 7, since 42 = 6 × 7. A number is divisible by 6 if it is divisible by both 2 and 3. Step 1: Check divisibility by 2 for each option. A number is divisible by 2 if its last digit is even. (A) 26565: Ends in 5 (odd). Not divisible by 2. (B) 26574: Ends in 4 (even). Potentially divisible by 2. (C) 26586: Ends in 6 (even). Potentially divisible by 2. (D) 26600: Ends in 0 (even). Potentially divisible by 2. Option (A) is eliminated. Step 2: Check divisibility by 3 for the remaining options. A number is divisible by 3 if the sum of its digits is divisible by 3. (B) 26574: Sum of digits = 2+6+5+7+4 = 24. Since 24 is divisible by 3, 26574 is divisible by 3. (Since it's also divisible by 2, it's divisible by 6). (C) 26586: Sum of digits = 2+6+5+8+6 = 27. Since 27 is divisible by 3, 26586 is divisible by 3. (Since it's also divisible by 2, it's divisible by 6). (D) 26600: Sum of digits = 2+6+6+0+0 = 14. Since 14 is not divisible by 3, 26600 is not divisible by 3. Option (D) is eliminated. Step 3: Check divisibility by 7 for the remaining options (B) and (C). We will perform long division for these two numbers. For option (B) 26574: (26574)/(42) Divide 265 by 42: 265 = 6 × 42 + 13. Bring down 7, making 137. Divide 137 by 42: 137 = 3 × 42 + 11. Bring down 4, making 114. Divide 114 by 42: 114 = 2 × 42 + 30. Since there is a remainder of 30, 26574 is not perfectly divisible by 42. For option (C) 26586: (26586)/(42) Divide 265 by 42: 265 = 6 × 42 + 13. Bring down 8, making 138. Divide 138 by 42: 138 = 3 × 42 + 12. Bring down 6, making 126. Divide 126 by 42: 126 = 3 × 42 + 0. Since the remainder is 0, 26586 is perfectly divisible by 42. Thus, 26586 is the number that, when divided by 42, yields no fraction. The final answer is C) 26586. That's 2 down. 3 left today — send the next one.