[3 marks sum]

Question 13 Marks

$5 \times 555$ a multiple of $9$?

Answer

Sum of the digits of $5 \times 555$
$= 5 + x + 5 + 5 + 5 = 20 + x$
It is multiple by $9$
The sum should be divisible by $9$
Value of $x$ will be $7$

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Question 23 Marks

$7\times 34$ divisible by $9$?

Answer

$7\times 34 $ is multiple of $9$
$\Rightarrow 7 + x + 3+ 4$ is a multiple of $9$
$\Rightarrow 14 + x = 0, 9, 18, 27,$
$\Rightarrow x = -1, 4, 13,$
Since,$ x$ is a digit
$x = 4$

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Question 33 Marks

$9142 \times a$ multiple of $4$?

Answer

$9142 \times$ is multiple of $4$
$\Rightarrow 9 + 1 + 4 + 2 + x $ is a multiple of $4$
$\Rightarrow 16 + x = 0, 4, 8, ………$
$x = -8, -4, 0, 4, 8$
Since, $x$ is $a$ digit
$4, 8$

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Question 43 Marks

$3\times 26$ a multiple of $6$?

Answer

$3\times 26$ is a multiple of $6$
$3 + x + 2 + 6$ is a multiple of $3$
$\Rightarrow 11 + x = 0, 3, 6, 9, 12, 15, 18,21,$
$\Rightarrow x = -11, -8, -5, -2, 1, 4, 7, 10, ….$
Since, $x$ is $a $digit
$x = 1, 4$ or $7$

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Question 53 Marks

$24x$ divisible by $ 6$?

Answer

$24x$ is divisible by $6$
$\Rightarrow 2 + 4+ x$ is a multiple of $6$
$\Rightarrow 6 + x = 0, 6, 12$
$\Rightarrow x = -6, 0, 6$
Since,$ x$ is a digit
$x = 0, 6$

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Question 63 Marks

$31×5$ divisible by $3$?

Answer

$31\times 5$ is divisible by $3$
$\Rightarrow 3 + 1 + x + 5$ is a multiple of $3$
$\Rightarrow 9 + x = 0, 3, 6, 9,$
$\Rightarrow x = -9, -6, -3, 0, 3, 6, 9,$
Since, $x$ is a digit
$x = 0, 3, 6$ or $9$

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Question 73 Marks

$1×5$ divisible by $3$?

Answer

$1\times 5$ is divisible by $3$
$\Rightarrow 1 + x + 5$ is a multiple of $3$
$\Rightarrow 6 + x = 0, 3, 6, 9,$
$\Rightarrow x = -6, -3, 0, 3, 6, 9$
Since, $x $is a digit
$x = 0, 3, 6$ or $9$

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Question 83 Marks

$5 × 2$ a multiple of $11$?

Answer

Sum of a digit in even place $= x$
and sum of the digits in odd place $= 5 + 2 = 7$
Difference of the sum of the digits in even places and in odd places $= x – 7$
$5\times 2$ is a multiple of $11$
$\Rightarrow x – 7 = 0, 11, 22,$
$\Rightarrow x = 7, 18, 29,$
Since, $x$ is a digit
$x = 7$

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Question 93 Marks

$3×2$ divisible by $11$?

Answer

Sum of the digit in even place $= x$
and sum of the digits in odd place $= 3 + 2 = 5$
Difference of the sum of the digits in even places and in odd places $= x – 5$
$3\times 2$ is a multiple of $11$
$\Rightarrow x – 5 = 0, 11, 22,$
$\Rightarrow x = 5, 16, 27,$
Since,$x$ is a digit $x = 5$

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Question 103 Marks

$A B\times5C A B$

Answer

As we need $\text{B}$ at unit place and $\text{A}$ at ten’s place,
$B = 0$ as $5 \times 0 = 0$
Now we want to find $\text{A}, 5 \times \text{A = A}($at unit’s place$)$
$\text{A} = 5,$ as $ 5 \times 5 = 25$
$\text{C} = 2$
$50\times 5=2 5 0$

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Question 113 Marks

$ AB \times3CAB$

Answer

As we need $B$ at unit place and $A $ at ten's places,
$\therefore B = 0$ as $3 \times 0 = 0$
Now we want to find $A, 3 \times A = A ($at unit's place$)$
$\therefore A = 5, as \ 3 \times 5 = 15$
$\therefore C = 1$
$50\times3=150$

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Question 123 Marks

$\text{AB}\times 6\text{BBB}$

Answer

As we need $\text{B}$ at unit place and $\text{B}$ at ten's place,
$∴\text{B} = 4$ as $6 \times 4 = 24$
Now we want to find $\text{A}, 6 \times\text{A} + 2 = 4 ($at unit's place$)$
$∴ \text{A} = 7$
$74\times 6=444$

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Question 133 Marks

$1\text{A}\times \text{A}9\text{A}$

Answer

As we need $\text{A}$ at unit place and $9$ at ten’s place,
$\text{A} = 6$ as $6 \times 6 = 36$
$16\times 6=96$

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Question 143 Marks

$12\text{A}+6\text{B}=\text{A}09$

Answer

$A + B = 9$
and $2 + A = 10$
$\therefore A = 10 - 2 = 8$
and $8 + B = 9$
$\therefore B = 9 - 8 = 1$
Hence $A = 8$ and $B = 1$
$128+681=809$

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Question 153 Marks

$2AB+ AB 1=B18$

Answer

$B = 7$ as $7 + 1 = 8.$
We want $8$ at unit place.
Now
$7 + A = 11$
$\therefore A = 11 - 7 = 4$
Hence $A = 4$ and $B = 7$
$247+471=718$

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Question 163 Marks

$A1+1B=B0$

Answer

$B = 9$ as $9 + 1 = 10.$
We want $0$ at units place and $1$ is carry over.
Now $B - 1 -1 = A.$
$\therefore A = 9 - 2 = 7$
Hence$ A + 7$ and $B = 9$
$71+19=90$

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Question 173 Marks

$98+4A=CB3$

Answer

$A = 5$ as $8 + 5 = 13$.
We want $3$ at units place and $1$ is carry over.
Now $9 + 4 + 1 = 14.$
$B = 4$ and $C = 1$
Hence $A = 5$ and $B = 4$ and $C = 1$
$98+ 45=143$

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Question 183 Marks

$3A+ 25=B2$

Answer

$A = 7$ as $7 + 5 = 12.$
We want $2$ at units place and $1$ is carry over.
Now $3 + 2 + 1 = 6$
$B = 6$
Hence $A = 7$ and $B = 6$
$37+25=62$

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Question 193 Marks

$6AB5+ D58C=9351$

Answer

$C + 5 = 11$
$\therefore C = 11 - 5 = 6$
and $8 + B + 1 = 15$
$\therefore B = 15 -9 = 6$
and $A + 5 + 1 = 13$
$\therefore A = 13 - 6 = 7$
and $6 + D + 1 = 9$
$\therefore D = 9 - 7 = 2$
Hence $A = 7, B = 6, C = 6$ and $D = 2$
$6765+2586=9351$

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Question 203 Marks

$8A5+94A=1A33$

Answer

$5 + A = 13$
and $A + 4 = 13$
$\therefore A = 13 - 5 = 8$
Hence $A = 8$
$885+948=1833$

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Question 213 Marks

If $c > a$; show that $cba – abc = 99(c – a).$

Answer

Given : $ c > a$
To show : $cba – abc = 99(c – a)$
Proof:
$cba = 100c + 106 + a ……….(i)$
$($By using property $3)$
$abc = 100a + 106 + c ………(ii)$
$($By using property $3)$
Subtracting $(ii)$ from$ (i)$
$cba – abc= 100c+ 106 + a- 100a- 106-c$
$\Rightarrow cba – abc = 99c – 99a$
$\Rightarrow cba – abc = 99(c – a)$
Hence proved.

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Question 223 Marks

Find the quotient when $94 – 49$ is divided by $5$

Answer

Difference of $94$ and $49$ is to be divided by $5$
Let $ab = 94$ and $ba = 49$
$\therefore a = 9$ and $b = 4$
The quotient of $94 - 49\ i.e. (ab -ba)$ when divided by $5\ i.e. (a -b)\ is\ 9$
$\left(\because \frac{a b-b a}{a-b}=9\right)$

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Question 233 Marks

Find the quotient when $94 – 49$ is divided by $9$

Answer

Difference of $94$ and $49$ is to be divided by $9$
Let $ab = 94$ and $ba = 49$
$\therefore a = 9$ and $b = 4$
The quotient of $94 - 49$ i.e. $(ab - ba)$
when divided by $9$ is $(a -b)$ i.e. $9 - 4 = 5$
$\left(\because \frac{a b-b a}{9}=a-b\right)$

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Question 243 Marks

Find the quotient when $73 – 37$ is divided by $4.$

Answer

Difference of $73 - 37$ is to be divided by $4$
Let $ab = 73$ and $ba = 37$
$\therefore a = 7$ and $b = 3$
The quotient of $73 - 37$ i.e. $(ab - ba)$ when divided by $4$ i.e. $(a -b)$ is $9$
$\left(\because \frac{a b-b a}{a-b}=9\right)$

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Question 253 Marks

Find the quotient when $73 – 37$ is divided by $9.$

Answer

Difference of $73 - 37$ is to be divided by $9$
Let $ab = 73$ and $ba = 37$
$\therefore a = 7$ and $b = 3$
The quotient of $73 - 37 ($i.e. $ab - bc)$ when divided by $7$ is $a - b$ i.e. $7 - 3 = 4$
$\left(\because \frac{a b-b a}{9}=a-b\right)$

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Question 263 Marks

Write the quotient when the sum of $94$ and $49$ is divided by $13.$

Answer

Sum of $94$ and $49$ is to be divided by $13$
Let $ab = 94$ and $ba = 49$
$\therefore a = 9$ and $b = 4$
The quotient of $94 + 49 ($i.e. $ab + ba)$
When divided by $13$ i.e. $(a + b)$ is $11$
$\left(\because \frac{a b+b a}{a+b}=11\right)$

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Question 273 Marks

Write the quotient when the sum of $94$ and $49$ is divided by $11$.

Answer

Sum of $94$ and $49$ is to be divided by $11$
Let $ab = 94$
and $ ba = 49$
$\therefore a = 9$ and $b = 4$
The quotient of $94 + 49 ($i.e. $ab + ba)$
When divided by $11$ is $a + b$ i.e. $9 + 4 = 13$
$\left(\because \frac{a b+b a}{11}=a+b\right)$

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Question 283 Marks

Write the quotient when the sum of $73$ and $37$ is divided by $10$.

Answer

Sum of $73$ and $37$ is to be divided by $10$
Let $ab = 73$
and $ba = 37$
$\therefore a = 7$
and $b = 3$
The quotient of $ab + ba$ i.e$. (73 + 37)$ when divided by $10 ($i.e. $a + b)$ is $11$
$\left(\because \frac{a b+b a}{a+b}=11\right)$

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Question 293 Marks

Write the quotient when the sum of $73$ and $37$ is divided by $11$.

Answer

Sum of $73$ and $37$ is to be divided by $11$
Let $ab = 73$
and $ba = 37$
$\therefore a = 7$
and $b = 3$
The quotient of $ab + bc$ i.e. $(73 + 37)$ when divided by $11$ is $a + b = 7 + 3 = 10$
$\left(\because \frac{a b+b a}{11}=a+b\right)$

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MATHS [3 marks sum]

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