2 Marks Questions

Question 12 Marks

Given an example of a number which is divisible by: $3$ but not by $6.$

Answer

$15$. Every number with the structure $(6n + 3)$ is an example of a number that is divisible by $3$ but not by $6.$

Question 22 Marks

Find the remainder when $981547$ is divided by $5.$ Do this without doing actual division.

Answer

A number is divisible by $5$ if its units digit is $0$ or $5$ But in number $981547,$ units digit is $7$
$\therefore$ Dividing the number by $5,$ Then remainder will be $7 - 5 = 2$

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Question 32 Marks

Show that the Cryptarithm $4\times\overline{\text{AB}}=\overline{\text{CAB}}$ does not have any solution.

Answer

$0$ is the only unit digit number, which gives the same $0$ at the unit digit when multipied by $4.$ So, the possible value of $B$ is $0.$ Similarly, for $A, 0$ is the only possible digit.But then $A, B$ and $C$ will all be $0.$ And if $A, B$ and $C$ become $0,$ these numbers cannot be of two-digit or three-digit. Therefore, both will become a one-digit number.Thus, there is no solution possible.

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Question 42 Marks

Given an example of a number which is divisible by: $4$ but not by $8.$

Answer

$28$. Every number with the structure $(8n + 4)$ is an example of a number that is divisible by $4$ but not by $8.$

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Question 52 Marks

Given an example of a number which is divisible by: $2$ but not by $4.$

Answer

$10.$ Every number with the structure $(4n + 2)$ is an example of a number that is divisible by $2$ but not by $4.$

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Question 62 Marks

Find the remainder when $51439786$ is divided by $3.$ Do this without performing actual division.

Answer

In the number $51439786,$ sum of digits is $5 + 1 + 4 + 3 + 9 + 7 + 8 + 6 = 43$ and the given number is divided by $3.$
$\therefore$ The sum of digits must by divisible by $3$
$\therefore$ Divisible $43$ by $3,$ the remainder will be $= 1$ Hence remainder $= 1$

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Question 72 Marks

If $\overline{98215\text{x}2}$ is a number with $x$ as its tens digit such that is is divisible by $4.$ Find all possible values of $x.$

Answer

A natural number is divisible by $4$ if the number formed by its digits in units and tens place is divisible by $4.$
$\therefore\overline{98215\text{x}2}$ will be divisible by $4$ if $\overline{\text{x}2}$ is divisible by $4.$
$\therefore\overline{\text{x}2}=10\text{x}+2$ x is a digit; therefore possible values of $x$ are $0, 1, 2, 3, 9.$
$\overline{\text{x}2}$
$= 2, 12, 22, 32, 42, 52, 62, 72, 82, 92$ The numbers that are divisible by $4$ are $12, 32, 52, 72, 92.$ Therefore, the values of $x$ are $1, 3, 5, 7, 9.$

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Question 82 Marks

Given an example of a number which is divisible by: both $4$ and $8$ but not by $32.$

Answer

$8$ Every number with the structure $(32n + 8), (32n + 16)$ or $(32n + 24)$ is an example of a number that is divisible by $4$ and $8$ but not by $32.$

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Question 92 Marks

Without performing actual division, find the remainder when $928174653$ is divided by $11.$

Answer

Let $n = 928174653 =$ A multiple of $11 + (9 + 8 + 7 + 6 + 3) - (2 + 1 + 4 + 5) =$ A multiple of $11 + 33 - 12 =$ A multiple of $11 + 21 =$ A multiple of $11 + 11 + 10 =$ A multiple of $11 + 10$
$\therefore$ Remainder $= 10$

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