Express 3 780 as the product of its prime factors in index notation. Answer 3 780 = ... The lowest common multiple of 3 780 and integer k is 7 560.
The highest common factor of 3 780 and integer k is 60. Find the smallest possible integer value of k. Answer k = ...

1. Break 3780 down into prime factors and collect repeated factors.

$3780=378\times 10$
$= (2\times 3^3\times 7)(2\times 5)$
$=2^2\times 3^3\times 5\times 7$

1. Write the given numbers in prime factor form.

From part (a),
$3780 = 2^2 \times 3^3 \times 5 \times 7$.

Also,
$7560 = 2 \times 3780 = 2^3 \times 3^3 \times 5 \times 7$,

and
$60 = 2^2 \times 3 \times 5$.

2. Determine the prime powers in the smallest possible value of $k$.

Let the exponents of $2,3,5,7$ in $k$ be chosen so that:

• HCF with $3780$ is $2^2 \times 3^1 \times 5^1$,
• LCM with $3780$ is $2^3 \times 3^3 \times 5^1 \times 7^1$.

So $k$ must contain:

$2^3$ to make the LCM have $2^3$,

$3^1$ and $5^1$ to make the HCF include $3$ and $5$,

no factor $7$ to make the HCF exclude $7$.

Thus the smallest possible
$k = 2^3 \times 3 \times 5 = 120$.