If $525 \equiv (10 + K)( {mod } 7)$ where $K \in \mathbb{N}$, then the least value of $K$ is:

We need to find the smallest natural number $K$ that satisfies:

$525 \equiv (10 + K) \pmod{7}$

When 525 is divided by 7, the remainder should equal the remainder when $(10 + K)$ is divided by 7.

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Divide 525 by 7:

$525 \div 7 = 75$ exactly

$7 \times 75 = 525$

Therefore, $525 \equiv 0 \pmod{7}$

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Divide 10 by 7:

$10 \div 7 = 1$ remainder $3$

$7 \times 1 = 7$ and $10 - 7 = 3$

Therefore, $10 \equiv 3 \pmod{7}$

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The left side gives remainder 0.

The right side is $(10 + K) \equiv (3 + K) \pmod{7}$

For both sides to be equal:

$3 + K \equiv 0 \pmod{7}$

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The value $3 + K$ must be a multiple of 7.

The smallest multiple of 7 greater than 3 is 7 itself.

$3 + K = 7$

$K = 4$

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The least value of $K$ is $4$.