Answers

Short answers

5. LCM

• a) \(\boxed{408}\)
• b) \(\boxed{1260}\)

6. HCF

• a) \(\boxed{60}\)
• b) \(\boxed{54}\)

7. The first common time is \(\boxed{39\text{ years}}\).

Key steps

5a. \(\operatorname{LCM}(24,68)\)

\[ 24=2^3\cdot3,\qquad 68=2^2\cdot17 \]

Take the greatest power of every prime:

\[ \operatorname{LCM}=2^3\cdot3\cdot17=408 \]

Check:

\[ 408\div24=17,\qquad 408\div68=6 \]

5b. \(\operatorname{LCM}(420,180)\)

\[ 420=2^2\cdot3\cdot5\cdot7 \]

\[ 180=2^2\cdot3^2\cdot5 \]

\[ \operatorname{LCM}=2^2\cdot3^2\cdot5\cdot7=1260 \]

6a. \(\operatorname{HCF}(180,300)\)

\[ 180=2^2\cdot3^2\cdot5,\qquad 300=2^2\cdot3\cdot5^2 \]

Take the common prime factors with the smallest powers:

\[ \operatorname{HCF}=2^2\cdot3\cdot5=60 \]

6b. \(\operatorname{HCF}(270,378)\)

\[ 270=2\cdot3^3\cdot5,\qquad 378=2\cdot3^3\cdot7 \]

\[ \operatorname{HCF}=2\cdot3^3=54 \]

7. Cicadas and predators

The cycles are \(13\) years and \(3\) years. Their first common multiple is:

\[ \operatorname{LCM}(13,3)=13\cdot3=39 \]

So they coincide again after 39 years. The actual question after the scenario is cut off; please upload the lower part to check the exact wording.