Find the least number which when divided by 16, 18, 20 and 25 leaves 4 as remainder in each case, but when divided by 7 leaves no remainder.

### Concept & Formula To find a number that leaves a remainder $r$ when divided by multiple divisors $a, b, c$, we use the Least Common Multiple (LCM): $$\\text{Number} = \\text{LCM}(a, b, c) \\times k + r$$ where $k$ is a positive integer chosen such that the condition for the secondary divisor is fully satisfied. ### Step-by-Step Solution **Given:** * Divisors: 16, 18, 20, 25 * Remainder in each case: 4 * Complete divisibility by: 7 **Calculation:** 1. Find the LCM of 16, 18, 20, and 25: * $16 = 2^4$ * $18 = 2 \\times 3^2$ * $20 = 2^2 \\times 5$ * $25 = 5^2$ * $\\text{LCM} = 2^4 \\times 3^2 \\times 5^2 = 16 \\times 9 \\times 25 = 3600$ 2. Express the required number in the standard form: $$\\text{Number} = 3600k + 4$$ 3. Test values of $k$ such that $(3600k + 4)$ is perfectly divisible by 7: * Simplify $3600 \\pmod 7$: $3600 = 7 \\times 514 + 2$ * So, the expression becomes: $2k + 4$ * For $k = 1 \\implies 2(1) + 4 = 6$ (not divisible by 7) * For $k = 2 \\implies 2(2) + 4 = 8$ (not divisible by 7) * For $k = 3 \\implies 2(3) + 4 = 10$ (not divisible by 7) * For $k = 4 \\implies 2(4) + 4 = 12$ (not divisible by 7) * For $k = 5 \\implies 2(5) + 4 = 14$ (divisible by 7) 4. Substitute $k = 5$ into the original expression: $$\\text{Number} = 3600 \\times 5 + 4 = 18000 + 4 = 18004$$ ### Exam Strategy & Shortcut **Option Elimination:** Check the options for divisibility by 7 directly. * 17004: $17004 \\div 7 = 2429.14$ (No) * 18000: Not possible since it leaves a remainder of 0 when divided by 20 and 25, not 4. * 18002: $18002 \\div 7 = 2571.71$ (No) * 18004: $18004 \\div 7 = 2572$ (Yes) Only **18004** satisfies the divisibility condition for 7. ### Common Pitfall Students often forget to find the specific value of $k$ and mark a value like $3600 + 4 = 3604$ if it is in the options, ignoring the divisibility test for the number 7. Always verify the secondary divisibility condition. ### Final Answer **Therefore, the correct answer is 18004.**