2 Mark Question

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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