Question paper, Set 6

Natural numbers are the counting numbers $1, 2, 3, 4, \ldots$

From $118$ to $123$ includes both $118$ and $123$.

$118, 119, 120, 121, 122, 123$

Answer: $118, 119, 120, 121, 122, 123$

Odd numbers give remainder $1$ when divided by $2$: $1, 3, 5, 7, \ldots$

First five: $1, 3, 5, 7, 9$

Answer: $1, 3, 5, 7, 9$

Multiples of $14$: $14 \times 1, 14 \times 2, 14 \times 3, \ldots$

$14 \times 1 = 14$, $14 \times 2 = 28$, $14 \times 3 = 42$, $14 \times 4 = 56$

$14 \times 5 = 70$, $14 \times 6 = 84$, $14 \times 7 = 98$

$14 \times 8 = 112$ is not less than $100$

Multiples of $14$ less than $100$: $14, 28, 42, 56, 70, 84, 98$

Answer: $14, 28, 42, 56, 70, 84, 98$

$27 = 1 \times 27$

$= 3 \times 9$

$\therefore$ Factors of $27$ are $1, 3, 9, 27$

Answer: $1, 3, 9, 27$

$140 = 1 \times 140$

$= 2 \times 70$

$= 4 \times 35$

$= 5 \times 28$

$= 7 \times 20$

$= 10 \times 14$

$3, 6, 8, 9, 11$ do not divide $140$ (they leave a remainder).

$\therefore$ Divisors of $140$ are $1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140$

Number of divisors $= 12$

Answer: $1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140$ ($12$ divisors)

$t$ can take various numerical values.

So $t$ is a variable.

Answer: Variable

As $-2 \times a = -2a$:

$-3 \times n = -3n$

Answer: $-3n$

(i) $7c + 8c = (7 + 8)c = 15c$

(ii) $20c - 11c = (20 - 11)c = 9c$

Answer: (i) $15c$ (ii) $9c$

$x$ is multiplied $2$ times.

$x \times x = x^{2}$

Read it as '$x$ power $2$' (or) '$x$ squared'.

Answer: $x^{2}$, read as '$x$ power $2$' (or) '$x$ squared'

$65 \times 5 = 325$

$65 \times 6 = 390$, written one place to the left: $3900$

$325 + 3900 = 4225$

$\therefore 65 \times 65 = 4225$

Answer: $4225$

$13^{3} = 13 \times 13 \times 13$

$13 \times 13 = 169$, $169 \times 13 = 2197$

$\therefore 13^{3} = 2197$

Answer: $2197$

(i) Divide by $11$ again and again:

$14641 \div 11 = 1331$, $1331 \div 11 = 121$, $121 \div 11 = 11$

$14641 = 11 \times 11 \times 11 \times 11 = 11^{4}$

(ii) Divide by $121$ again and again:

$14641 \div 121 = 121$

$14641 = 121 \times 121 = 121^{2}$

Answer: (i) $11^{4}$ (ii) $121^{2}$

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 $73120$ is $0$, so $73120$ is divisible by $2$.

Answer: Yes, the unit digit is $0$

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

Sum of the digits of $913254$: $9 + 1 + 3 + 2 + 5 + 4 = 24$

$24$ is divisible by $3$ ($24 = 3 \times 8$), so $913254$ is divisible by $3$.

Answer: Yes, the sum of the digits is $24$

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

$1309$: $1 + 3 + 0 + 9 = 13$, not divisible by $3$

$1308$: $1 + 3 + 0 + 8 = 12$, divisible by $3$

$\therefore$ The largest number less than $1310$ that is divisible by $3$ is $1308$.

Answer: $1308$

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

Last two digits of $13562$ together: $62$; $62 = 4 \times 15 + 2$, so $13562$ is not divisible by $4$.

Answer: No, the last two digits $62$ are not 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 $50557$ is $7$, so $50557$ is not divisible by $5$.

Answer: No, the unit digit is $7$

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

The digit at the unit place of $61254$ is $4$, so $61254$ is divisible by $2$.

Sum of the digits of $61254$: $6 + 1 + 2 + 5 + 4 = 18$

$18$ is divisible by $3$ ($18 = 3 \times 6$), so $61254$ is divisible by $3$.

$\therefore$ $61254$ is divisible by $6$.

Answer: Yes, it is divisible by both $2$ and $3$

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

$749$: $74 - 9(2) = 74 - 18 = 56$; $56 = 7 \times 8$, a multiple of $7$, so $749$ is divisible by $7$.

$994$: $99 - 4(2) = 99 - 8 = 91$; $91 = 7 \times 13$, a multiple of $7$, so $994$ is divisible by $7$.

$426$: $42 - 6(2) = 42 - 12 = 30$; $30 = 7 \times 4 + 2$, not a multiple of $7$, so $426$ is not divisible by $7$.

Answer: $749$ and $994$ are divisible by $7$; $426$ is not.

By $2$: The digit at the unit place of $26472$ is $2$, so $26472$ is divisible by $2$.

By $3$: Sum of the digits of $26472$: $2 + 6 + 4 + 7 + 2 = 21$; $21$ is divisible by $3$ ($21 = 3 \times 7$), so $26472$ is divisible by $3$.

By $4$: Last two digits of $26472$ together: $72$; $72 = 4 \times 18$, so $26472$ is divisible by $4$.

By $5$: The digit at the unit place of $26472$ is $2$, so $26472$ is not divisible by $5$.

By $6$: $26472$ is divisible by both $2$ and $3$, so it is divisible by $6$.

$\therefore$ $26472$ is divisible by $2, 3, 4$ and $6$.

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