Question

Find the least number by which 14450 number should be multiplied to make them perfect square

Answer 1

Answer:

Find the smallest number by which 14,450 should be multiplied to make it a perfect square. Not perfect square. therefore,if we multiply it by 2 it will become a perfect square. Multiplied by 2 so it make a perfect square.

Answer 2

We have,

• To figure out number which is should be multiplied with 14450 to obtain a perfect square number.

Concept :

When the prime factors of a number has even exponential value, the number is a perfect square number. It can be understood by an example --

$\rm{}xyz = {x}^{n} \times {y}^{m} \times {z}^{r}$

For above, n,m, and r should be even, so that xyz could be a perfect square number.

SOLUTION :::

♦ Prime factorisation of 14450

$\underline{ \: \: \: \: 2|14450} \\ \underline{ \: \: \: \: 5|7225 \: \: } \\ \underline{ \: \: \: \: 5| 1445\: \: } \\ \underline{ 17| 289\: \: \:} \\ \underline{ \: \: \: \: | 17\: \: \: \: }$

$\tt\therefore \: \: 14450 = \purple{2 \times {5}^{2} \times {17}^{2} }$

Here,

• Only the 2 has non-even exponent i.e. 2 is not in pair of two.
• So, 2 should be multiplied with factors to get a pair of two (or say even exponent of 2)

Thus,

• 2 is that required which on multiplying with 14450, will result a perfect square number.

Now,

$\tt \large \: 2 \times 14450 \\ \to \tt28900$

** Checking whether the obtained result is a perfect square number or not --

$\sf \huge \sqrt{28900} \\ \\ \sf \to \sqrt{2 \times 2 \times 5 \times 5 \times 17 \times 17} \\\to \sf \sqrt{ {2}^{2} \times {5}^{2} \times {17}^{2} } \\ \to \sf \sqrt{ {2}^{2} } \times \sqrt{ {5}^{2} } \times \sqrt{ {17}^{2} } \\\to \sf 2 \times 5 \times 17 = \pink{170}$

Hence, It is concluded that on multiplying 2 to 14450, we obtain a perfect square number.