3 Marks Question

Question 513 Marks

Define:
Factor

Answer

Factor:A factor of a number is an exact divisor of that number.
For example, $4$ exactly divide $32$. Therefore, $4$ is a factor of $32.$
Examples of factors are:
$2$ and $3$ are factors of $6$ because $2 \times 3 = 6$
$2$ and $4$ are factors of $8$ because $2 \times 4 = 8$
$3$ and $4$ are factors of $12$ because $3 \times 4 = 12$
$3$ and $5$ are factors of $15$ because $3 \times 5 = 15$

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Question 523 Marks

A rectangular courtyard is $20m$ $16\ cm$ long and $15m$ $60\ cm$ broad. It is to be paved with square stones of the same size. Find the least possible number of such stones.

Answer

Length of the rectangular courtyard $= 20m$
$16\ cm = 2,016\ cm$
Breadth of the rectangular courtyard $= 15m$
$60\ cm = 1,560\ cm$
Least possible side of the square stones used to pave the rectangular courtyard
$= HCF$ of $(2,016$ and $1,560)$
Prime factorization of $2,016 =2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7$
Prime factorization of $1,560 = 2 \times 2 \times 2 \times 3 \times 5 \times 13$
$HCF$ of $(2,016, 1,560) = 2 \times 2 \times 2 \times 3= 24$
Least possible side of square stones used to pave the rectangular courtyard is $24 \ cm.$
Number of square stones used to pave the rectangular courtyard
= Area of rectangular courtyard Area of square stone $= 2016\ cm \times 1560\ cm (24\ cm) 2 = 5460$
Thus, the least number of square stones used to pave the rectangular courtyard is $5,460.$

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Question 533 Marks

The length, breadth and height of a room are $8m$ $25\ cm, 6m$ $75\ cm$ and $4m$ $50\ cm$, respectively. Determine the longest rod which can measure the three dimensions if the room exactly.

Answer

Length of the room $= 8m$ $25\ cm = 825\ cm$
Breadth of the room $= 6m$ $75\ cm = 675\ cm$
Height of the room $= 4m$ $50\ cm = 450\ cm$
The longest rod will be given by the $HCF$ of $825, 675$ and $450.$
Prime factorization of $825 = 3 \times 5 \times 5 \times 11$
Prime factorization of $675 = 3 \times 3 \times 3 \times 5 \times 5$
Prime factorization of $450 = 2 \times 3 \times 3 \times 5 \times 5$
Therefore, $HCF$ of $825, 675$ and $450 = 3 \times 5 \times 5 = 75$
Thus, the required length of the longest rod is $75\ cm.$

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Question 543 Marks

Find the greatest number of four digits which is exactly divisible by each of $8, 12, 18$ and $30.$

Answer

$ 8=1 \times 2 \times 2 \times 2=2^3 $
$ 12=1 \times 2 \times 2 \times 3=2^2 \times 3^1$
$ 18=1 \times 2 \times 3 \times 3=2^1 \times 3^2 $
$ 30=1 \times 2 \times 3 \times 5=2^1 \times 3^1 \times 5^1$
$LCM$ of $8,12,18$, and $30=2^3 \times 3^2 \times 5^1=360$
Largest $4$-digit number is $9999$
Now, if we divide $9999$ by $360$, we will get $27.78$ as quotient.
The integer just less than $27.78$ is $27$
$\therefore$ Required number $= 360 \times 27 = 9720$
Hence, the greatest number of four digits which is exactly divisible by each of $8, 12, 18$ and $30$ is $9720$.

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MATHS 3 Marks Question