Answer:
k = 33
Step-by-step explanation:
Terminating decimal numbers: Decimals that have a finite number of decimal places.
For a decimal to be terminating, the factors of the denominator must only contain 2 and/or 5. As 2 and 5 are prime numbers, use prime factorization to rewrite the denominator.
Prime factorization of 660:
⇒ 660 = 2 × 2 × 3 × 5 × 11
⇒ 660 = 2² × 3 × 5 × 11
Therefore:
[tex]\implies \sf \dfrac{k}{660}=\dfrac{k}{2^2 \cdot 3 \cdot 5 \cdot 11}[/tex]
The fraction will only be a terminating decimal if both 3 and 11 in the denominator are canceled out. To do this, their lowest common multiple must be the numerator:
⇒ LCM of 3 and 11 = 3 × 11 = 33
[tex]\implies \sf \dfrac{33}{660}[/tex]
[tex]\implies \sf k=33[/tex]
Therefore, the smallest positive integer k such that k/660 can be expressed as a terminating decimal is 33.
