Question 15 Marks
Find the least $5-$digit number which is exactly divisible by $20, 25, 30.$
AnswerLeast five digit number $= 10000$
$\begin{array}{c|c}5&20,25,30\\\hline2&4,5,6\\\hline2&2,5,3\\\hline3&1,5,3\\\hline5&1,5,1\\\hline&1,1,1\end{array}$
$LCM$ of $20, 25, 30$ is $300.$
But we want the least five digit number which is divisible by $20, 25, 30$.
So, we will multiply the $LCM$ by a number that makes it the least five digit number divisible by $20, 25, 30.$
$300 \times 31 = 9300$
$300 \times 32 = 9600$
$300 \times 33 = 9900$
$300 \times 34 = 10200$
So, the least five digit number divisible by $20, 25, 30$ is $10200.$
View full question & answer→Question 25 Marks
Find the $HCF$ of the numbers in the following using the division method: $754, 1508, 1972$
AnswerThe given numbers are $754, 1508$ and $1972.$ First we will find the $HCF$ of $754$ and $1508.$
So, the $HCF$ of $754$ and $1508$ is $754.$
Now, we will find the $HCF$ of $754$ and $1972.$
So, the $HCF$ of $754$ and $1972$ is $58.$
$\therefore$ The $HCF$ of $754, 1058$ and $1972$ is $58.$ View full question & answer→Question 35 Marks
Find the greatest number of five digits exactly divisble by $9, 12, 15, 18$ and $24.$
AnswerFirst, we will find the $LCM$ Of $9, 12, 15, 18$ and $24.$
$\begin{array}{c|c}2&9,12,15,18,24\\\hline2&9,6,15,9,12\\\hline2&9,3,15,9,6\\\hline3&9,3,15,9,3\\\hline3&3,1,5,3,1\\\hline5&1,1,5,1,1\\\hline&1,1,1,1,1\end{array}$
$\therefore LCM$ of the numbers $= 2^3\times 3^2\times 5 = 360$
The least six-digit number $= 100000$
The greatest five-digit number divisible by $360.$
Weill be the quatient of $\frac{100000}{360}$ multiplied by $360.$
So, the greatest five-digit number exactly divisible by the given numbers will be $360 \times 277 = 99720.$ View full question & answer→Question 45 Marks
Find the greatest number that will divide $445, 572$ and $699,$ leaving remainders $4, 5, 6$ respectively.
AnswerSince the respective remainders of $445, 572$ and $699$ are $4, 5$ and $6,$ we have to find the number which exactly devides $(445 - 4), (572 - 5)$ and $(696 - 6)$.So, the required number is the $HCF$ of $441, 567$ and $693.$
Firstly, we will find the $HCF$ of $441$ and $567.$
$\therefore HCF = 63$
Now, we will find the $HCF$ of $63$ and $693.$
$\therefore HCF = 63$
Hence, the required numbers is $63.$ View full question & answer→Question 55 Marks
Three different containers contain $403L, 434L$ and $465L$ of milk respectively. Find the capacity of a container which can measure the milk of all containers in an exact number of times.
AnswerThree different containers contain $403L, 434L$ and $465L$ of milk.The capacity of the comntainer that can measure the milk in an exact number of times will be given by the $HCF$ of $403, 434$ and $465.$
$\therefore HCF = 31$
Now, we will find the $HCF$ of $31$ and $465.$
$\therefore HCF = 31.$
Hence, the capacity of the required container is $31L.$ View full question & answer→Question 65 Marks
The circumferences of four wheels are $50\ cm, 60\ cm, 75\ cm$ and $100\ cm.$ They start moving simultaneously. what least distance should they cover so that each wheel makes a complete number of revolutions?
AnswerDistance covered by a wheel for one complete revolution = circumference of the wheel.
All the wheels will make complete numbers of revolutions when the distances covered by them equal to their $LCM.$
$\begin{array}{c|c}5&50,60,75,100\\\hline5&10,12,15,20\\\hline2&2,12,3,4\\\hline2&1,6,3,2\\\hline3&1,3,3,1\\\hline&1,1,1,1\end{array}$
Required least distance $= 5 \times 5 \times 2 \times 2 \times 3$
$= 25 \times 4 \times 3$
$= 300\ cm = 3m$
So each wheel will make a complete number of revolutions after travelling $3m.$
View full question & answer→Question 75 Marks
Show that the following pairs are co-primes: $512, 945$
AnswerGiven numbers are $512$ and $945.$
$\begin{array}{c|c}2&512\\\hline2&256\\\hline2&128\\\hline2&64\\\hline2&32\\\hline2&16\\\hline2&8\\\hline2&4\\\hline2&2\\\hline&1\end{array}$
$\begin{array}{c|c}3&315\\\hline3&105\\\hline5&35\\\hline7&7\\\hline&1\end{array}$
Now, $512=2\times2\times2\times2\times2\times2\times2\times2\times2=2^9$
$945=3\times3\times3\times5\times7=3^3\times5\times7$
Thus, the $HCF$ of $512$ and $945$ is $1.$
$\therefore 512$ and $945$ are co-primes.
View full question & answer→Question 85 Marks
There are $527$ apples, $646$ pears and $748$ oranges. These are to be arranged in heaps containing the same number of fruits. Find the greatest number of fruits. Find the greatest number of fruits possible in each heap. How many heaps are formed?
AnswerNumber of apples $= 527$
Number of pears $= 646$
Number of oranges $= 748$
The fruits are to be arranged in heaps containing the same number of fruits.
The greatest number of fruits posible in each heaps will be given by the $HCF$ of $527, 646$ and $748.$
Firstly, we will find the $HCF$ of $527$ and $646.$
$\therefore HCF$ of $527, 646$ and $748 = 17$
So, the greatest number of fruits in each heaps will be $17.$ View full question & answer→Question 95 Marks
Find the least number of five digits that is exactly divisible by $16, 18, 24$ and $30.$
Answer$LCM$ of $16, 18, 24$ and $30.$
$\begin{array}{c|c}2&16,18,24,30\\\hline2&8,9,12,15\\\hline2&4,9,6,15\\\hline2&2,9,3,15\\\hline3&1,9,3,15\\\hline3&1,3,1,5\\\hline5&1,1,1,5\\\hline&1,1,1,1\end{array}$
$LCM= 2^4× 3^2× 5 = 720$ We have to find the least five-digit number that is exactly divisible by $16, 18,$ and $30.$
But $LCM = 720$ is a three digit number.
The least five digit number $= 10000.$ Dividing $10000$ by $720,$ we get:
The greatest fout- digits number exactly divisible by $720 = 10000 - 640 = 9360$
So, the least five-digit number exactly divisible by $720 = 9360 + 720 = 10080$ View full question & answer→Question 105 Marks
A rectangular courtyard is $18m, 72\ cm$ long and $13m, 20\ cm$ board. It is to be paved with square tiles of the same size. Find the least possibele number of such tiles.
AnswerLength of the courtyard $= 18m, 72\ cm = 1872\ cm$
Breadth of the courtyard $= 13m, 20\ cm = 1320\ cm$
Now, maximum edge of the square tile is given by the $HCF$ of $1872\ cm$ and $1320\ cm.$
$HCF$ of $1872$ and $1320 = 24$
$\therefore$ maximum edge of the squre tile $= 24\ cm$
Required number of tiles $=\frac{\text{area of courtyard}}{\text{area of each square tile}}$
$=\frac{1872\times1320}{24\times24}$
$=4290$ View full question & answer→Question 115 Marks
Find the $HCF$ of the numbers in the following using the division method: $1794, 2346, 4761$
AnswerThe given numbers are $1794, 2346$ and $4761.$ First we will find the $HCF$ of $1794$ and $2346.$
So, the $HCF$ of $1794$ and $2346$ is $138.$
Now, we will find the $HCF$ of $138$ and $4761.$
So, the $HCF$ of $138$ and $4761$ is $69.$
$\therefore$ The $HCF$ of $1794, 2346$ and $4761$ is $69.$ View full question & answer→Question 125 Marks
Find the $HCF$ of the numbers in the following using the division method: $391, 425, 527$
AnswerThe given numbers are $391, 425$ and $527$ First we will find the $HCF$ of $391$ and $425.$
So, the $HCF$ of $391$ and $42$5 is $17.$
Now, we will find the $HCF$ of $17$ and $527.$
So, the $HCF$ of $17$ and $527$ is $17.$
$\therefore$ The $HCF$ of $391, 425$ and $527$ is $17.$ View full question & answer→Question 135 Marks
Find the $HCF$ of the numbers in the following using the division method $:658, 940, 1128$
AnswerThe given numbers are $658, 940$ and $1128.$ First we will find the $HCF$ of $658 $ and $940.$
Thus, the $HCF$ of $658$ and $940$ is $94.$
Now, we will find the $HCF$ of $94$ and $1128.$
Thus, the $HCF$ of $94 $ and $1128$ is $94.$
$\therefore$ The $HCF$ of $658, 9402$ and $1128$ is $94.$ View full question & answer→Question 145 Marks
Find the largest number which divides $630$ and $940$ leaving remainders $6$ and $4$ respectively.
AnswerSince $6$ and $4$ are the remainders, the number must exactly divide the following:
$630 - 6 = 624$ and $940 - 4 = 936$
$\begin{array}{c|c}3&642\\\hline2&204\\\hline2&104\\\hline2&52\\\hline2&26\\\hline13&13\\\hline&1\end{array}$ $\begin{array}{c|c}3&936\\\hline2&312\\\hline2&156\\\hline2&78\\\hline3&39\\\hline13&13\\\hline&1\end{array}$
$624 = 2 \times 2 \times 2 \times 2 \times 3 \times 13 $
$936 = 2 \times 2 \times 2 \times 3 \times 3 \times 13$
$HCF$ of $624$ and $936 = 8 \times 3 \times 13 = 312$
So, $312$ is the greatest number that divides $630$ and $940,$
leaving $6$ and $4 $ as the respective remainders.
View full question & answer→Question 155 Marks
Define $(i) FACTOR, \ (ii) MULTIPLE.$ Give five example of each.
AnswerFactor: A factor of a number is an exact divisor of that number.
Multiple: A multiple of a number is a number obtained by multiplying it by a natural number.
Example 1: We know that $15 = 1 \times 15$ and $15 = 3 \times 5$
$\therefore 1, 3, 5$ and $15$ are the factors of $15.$
In other words, we can say that $15$ is a multiple of $1, 3, 5$ and $15.$
Example 2: We know that $8 = 8 \times 1, 8 = 2 \times 4$ and $8 = 4 \times 2$
$\therefore 1, 2, 4$ and $8$ are the factors of $8.$
In other words, we can say that $8$ is a multiple of $1, 2, 4$ and $8.$
Example 3: We know that $30 = 30 \times 1, 30 = 5 \times 6$ and $30 = 6 \times 5$
$\therefore 1, 5, 6$ and $30$ are factors of $30.$
In other words, we can say that $30$ is a multiple of $1, 5, 6$ and $30.$
Example 4: We know that $20 = 20 \times 1, 20 = 4 \times 5$ and $20 = 5 \times 4$
$\therefore 1, 4, 5$ and $20$ are factors of $20.$
In other words, we can say that $20$ is a multiple of $1, 4, 5$ and $20.$
Example 5: We know that $10 = 10 \times 1, 10 = 2 \times 5$ and $10 = 5 \times 2$
$\therefore 1, 2, 5$ and $10$ are factors of $10.$
In other words, we can say that $10$ is a multiple of $1, 2, 5$ and $10.$
View full question & answer→