Question paper, Set 9

Whole numbers are the natural numbers along with $0$: $0, 1, 2, 3, \ldots$

First four: $0, 1, 2, 3$

Answer: $0, 1, 2, 3$

$4161 = 2 \times 2080 + 1$

Remainder is $1$, so $4161$ is an odd number.

Answer: Odd (remainder $1$ when divided by $2$)

Multiples of $25$: $25 \times 1, 25 \times 2, 25 \times 3, \ldots$

$25 \times 1 = 25$, $25 \times 2 = 50$, $25 \times 3 = 75$, $25 \times 4 = 100$

$25 \times 5 = 125$, $25 \times 6 = 150$, $25 \times 7 = 175$, $25 \times 8 = 200$

$25 \times 9 = 225$ is not less than $210$

Multiples of $25$ less than $210$: $25, 50, 75, 100, 125, 150, 175, 200$

Answer: $25, 50, 75, 100, 125, 150, 175, 200$

$75 = 1 \times 75$

$= 3 \times 25$

$= 5 \times 15$

$\therefore$ Factors of $75$ are $1, 3, 5, 15, 25, 75$

Answer: $1, 3, 5, 15, 25, 75$

$160 = 1 \times 160$

$= 2 \times 80$

$= 4 \times 40$

$= 5 \times 32$

$= 8 \times 20$

$= 10 \times 16$

$3, 6, 7, 9, 11, 12$ do not divide $160$ (they leave a remainder).

$\therefore$ Divisors of $160$ are $1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160$

Number of divisors $= 12$

Answer: $1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160$ ($12$ divisors)

A symbol which takes various numerical values is called a variable (or) literal.

So another name for a variable is a literal.

Answer: Literal

As $2 \times a = 2a$, a number times a letter is written without the $\times$ sign.

$15 \times q = 15q$

Answer: $15q$

(i) $6q + 4q + 7q = (6 + 4 + 7)q = 17q$

(ii) $25q - 17q = (25 - 17)q = 8q$

Answer: (i) $17q$ (ii) $8q$

$r^{5} = r \times r \times r \times r \times r$

Read it as '$r$ power $5$'.

Answer: $r \times r \times r \times r \times r$, read as '$r$ power $5$'

$58 \times 8 = 464$

$58 \times 5 = 290$, written one place to the left: $2900$

$464 + 2900 = 3364$

$\therefore 58 \times 58 = 3364$

Answer: $3364$

$14^{3} = 14 \times 14 \times 14$

$14 \times 14 = 196$, $196 \times 14 = 2744$

$\therefore 14^{3} = 2744$

Answer: $2744$

(i) $100000000$ is $1$ followed by $8$ zeros.

$100000000 = 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 \times 10 = 10^{8}$

(ii) Divide by $100$ again and again:

$100000000 \div 100 = 1000000$, $1000000 \div 100 = 10000$, $10000 \div 100 = 100$

$100000000 = 100 \times 100 \times 100 \times 100 = 100^{4}$

Answer: (i) $10^{8}$ (ii) $100^{4}$

A number is divisible by $2$ if the digit at its unit place is $0, 2, 4, 6$ or $8$.

The digit at the unit place of $27594$ is $4$, so $27594$ is divisible by $2$.

Answer: Yes, the unit digit is $4$

A number is divisible by $3$ if the sum of all its digits is divisible by $3$.

Sum of the digits of $555556$: $5 + 5 + 5 + 5 + 5 + 6 = 31$

$31$ is not divisible by $3$ ($31 = 3 \times 10 + 1$), so $555556$ is not divisible by $3$.

Answer: No, the sum of the digits is $31$

A number is divisible by $3$ if the sum of all its digits is divisible by $3$.

$931$: $9 + 3 + 1 = 13$, not divisible by $3$

$932$: $9 + 3 + 2 = 14$, not divisible by $3$

$933$: $9 + 3 + 3 = 15$, divisible by $3$

$\therefore$ The smallest number greater than $930$ that is divisible by $3$ is $933$.

Answer: $933$

A number is divisible by $4$ if its last two digits together are divisible by $4$.

Last two digits of $60884$ together: $84$; $84 = 4 \times 21$, so $60884$ is divisible by $4$.

Answer: Yes, the last two digits $84$ are divisible by $4$

A number is divisible by $5$ if the digit at its unit place is $0$ or $5$.

The digit at the unit place of $94105$ is $5$, so $94105$ is divisible by $5$.

Answer: Yes, the unit digit is $5$

A number is divisible by $6$ if it is divisible by both $2$ and $3$.

The digit at the unit place of $70245$ is $5$, so $70245$ is not divisible by $2$.

Sum of the digits of $70245$: $7 + 0 + 2 + 4 + 5 = 18$

$18$ is divisible by $3$ ($18 = 3 \times 6$), so $70245$ is divisible by $3$.

$\therefore$ $70245$ is not divisible by $6$.

Answer: No, it is not divisible by $2$

A number is divisible by $7$ if the remaining part minus twice the unit digit is a multiple of $7$ (or $0$).

$336$: $33 - 6(2) = 33 - 12 = 21$; $21 = 7 \times 3$, a multiple of $7$, so $336$ is divisible by $7$.

$916$: $91 - 6(2) = 91 - 12 = 79$; $79 = 7 \times 11 + 2$, not a multiple of $7$, so $916$ is not divisible by $7$.

$434$: $43 - 4(2) = 43 - 8 = 35$; $35 = 7 \times 5$, a multiple of $7$, so $434$ is divisible by $7$.

Answer: $336$ and $434$ are divisible by $7$; $916$ is not.

By $2$: The digit at the unit place of $65118$ is $8$, so $65118$ is divisible by $2$.

By $3$: Sum of the digits of $65118$: $6 + 5 + 1 + 1 + 8 = 21$; $21$ is divisible by $3$ ($21 = 3 \times 7$), so $65118$ is divisible by $3$.

By $4$: Last two digits of $65118$ together: $18$; $18 = 4 \times 4 + 2$, so $65118$ is not divisible by $4$.

By $5$: The digit at the unit place of $65118$ is $8$, so $65118$ is not divisible by $5$.

By $6$: $65118$ is divisible by both $2$ and $3$, so it is divisible by $6$.

$\therefore$ $65118$ is divisible by $2, 3$ and $6$.

Answer: Divisible by $2, 3$ and $6$; not by $4$, $5$