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Among the given numbers below, the smallest number which will be divided by 9, 10, 15 and 20, leaves the remainders 4, 5, 10, and 15, respectively





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Solution

Find the smallest number which when divided by 9, 10, 15 and 20 leaves remainders 4, 5, 10 and 15 respectively.

✅ Solution:

Let the number be x.

  • x \equiv 4 \mod 9 \Rightarrow x - 4 divisible by 9
  • x \equiv 5 \mod 10 \Rightarrow x - 5 divisible by 10
  • x \equiv 10 \mod 15 \Rightarrow x - 10 divisible by 15
  • x \equiv 15 \mod 20 \Rightarrow x - 15 divisible by 20

So, x + 5 is divisible by LCM of 9, 10, 15, 20

LCM = 2^2 \cdot 3^2 \cdot 5 = 180

x + 5 = 180 \times 2 = 360 \Rightarrow x = 355

? Final Answer: \boxed{355}



The vector \vec{A}=(2x+1)\hat{i}+(x^2-6y)\hat{j}+(xy^2+3z)\hat{k} is a





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Solution

Vector Field:

\vec{A} = (2x + 1)\hat{i} + (x^2 - 6y)\hat{j} + (xy^2 + 3z)\hat{k}

Divergence:

\nabla \cdot \vec{A} = 2 - 6 + 3 = -1 \neq 0

Not solenoidal ❌

Curl:

\nabla \times \vec{A} = (2xy)\hat{i} - (y^2)\hat{j} + (2x)\hat{k} \neq \vec{0}

Not conservative ❌

Final Answer:

\vec{A} is neither conservative nor solenoidal.

Vector Sink Field Analysis

Given vector field:

\vec{A} = (2x + 1)\hat{i} + (x^2 - 6y)\hat{j} + (xy^2 + 3z)\hat{k}

Divergence:

\nabla \cdot \vec{A} = 2 - 6 + 3 = -1

✅ Conclusion:

The divergence is negative at every point, so \vec{A} is a sink field.



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