Express 1500 as the product of its prime factors. Leave your answer in index notations. Answer ___ $p = 2^{m}\times 3^{2}\times n$, where $m$ and $n$ are positive integers.
Find the values of $m$ and $n$ if the HCF of 1500 and $p$ is 6, given that $100 < p < 150$. Answer m = ___ n = ___ In a race, Car A, Car B and Car C took 3 minutes, 6 minutes and 8 minutes respectively to complete one lap around the racing track.
If the race started at 7 pm, at what time will the cars be back at the starting point together again? Answer ___

1. Break 1500 down into prime factors and express repeated factors using indices.

$1500 = 15 \times 100$
$= (3 \times 5) \times (2^2 \times 5^2)$
$= 2^2 \times 3 \times 5^3$

1. Compare the power of 2 in 1500, in $p$, and in the given HCF.

From 17(a)(i),
$1500=2^2\times 3\times 5^3$.

Given
$p=2^m\times 3^2\times n$ and the HCF is $6=2\times 3$.

For the factor 2, the HCF uses the smaller power of 2 from $1500$ and $p$:
$\min(2,m)=1.$
Since $m$ is a positive integer, $m=1$.

1. Substitute $m=1$ and use the range of $p$ to find possible integer values of $n$, then check the HCF condition.

Since $m=1$,
$p=2^1\times 3^2\times n=18n.$
Given $100<p<150$,
$100<18n<150$
$5.56<n<8.33.$
So $n=6,7,$ or $8$.

Check with $1500=2^2\times 3\times 5^3$:

• If $n=6$, then $p=108$ and the HCF is more than $6$.
• If $n=7$, then $p=126=2\times 3^2\times 7$, so the HCF with $1500$ is $2\times 3=6$.
• If $n=8$, then $p=144$ and the HCF is more than $6$.

Therefore, $n=7.$

1. Find the first time interval when all three cars return to the starting point together.

The cars will be back at the starting point together after the LCM of $3$, $6$ and $8$ minutes.

$3=3,\quad 6=2\times3,\quad 8=2^3$

$\text{LCM}=2^3\times3=24$

So they meet again after $24$ minutes.

2. Add the common time interval to the starting time.

Race started at $7.00$ pm.

$7.00\text{ pm}+24\text{ minutes}=7.24\text{ pm}$