What number will you multiply by 1323 to make it a cube number?

Question

Homi · Accepted Answer

Understand the Question

For a number to be a perfect cube, every prime factor in its prime factorisation must appear in triplets (groups of $3$ , i.e., powers of $3$ ).
To find the smallest number to multiply by $1323$ , we find its prime factorisation and determine how many additional prime factors are required to complete all triplets.

Step 1 · Find Prime Factorisation of 1323

Dividing $1323$ successively by prime factors: $1323 \div 3 = 441$ $441 \div 3 = 147$ $147 \div 3 = 49$ $49 \div 7 = 7$

Therefore, the prime factorisation of $1323$ is:

\begin{aligned} 1323 &= 3 \times 3 \times 3 \times 7 \times 7 \\ &= 3^3 \times 7^2 \end{aligned}

Step 2 · Find the Required Multiplier

Grouping the prime factors into triplets:

The factor $3$ appears $3$ times ( $3 \times 3 \times 3$ ), forming a complete triplet ( $3^3$ ).
The factor $7$ appears only $2$ times ( $7 \times 7$ ), which does not form a complete triplet.

To complete the triplet of $7$ , we need one more $7$ .

$\text{Multiplier} = 7$

Step 3 · Verify the Result

Multiplying $1323$ by $7$ : $1323 \times 7 = 9261$

In prime factor form: $3^3 \times 7^2 \times 7 = 3^3 \times 7^3 = (3 \times 7)^3 = 21^3$

\begin{aligned} 21^3 &= 21 \times 21 \times 21 \\ &= 441 \times 21 \\ &= 9261 \end{aligned}

Thus, $9261$ is a perfect cube ( $21^3$ ).

Answer

$7$

Common Mistakes

Multiplying vs. Dividing: Multiplying requires adding missing factors to complete triplets (multiply by $7$ ), whereas dividing requires removing unpaired factors (divide by $7^2 = 49$ ).
Over-multiplying: Multiplying by $7^2$ instead of $7$ . Only one additional factor of $7$ is needed to turn $7^2$ into $7^3$ .

Question 11