Question paper, Set 4

Natural numbers are the counting numbers $1, 2, 3, 4, \ldots$, so the smallest natural number is $1$.

Whole numbers are the natural numbers along with $0$: $0, 1, 2, 3, \ldots$, so the smallest whole number is $0$.

Answer: Smallest natural number $= 1$; smallest whole number $= 0$

$2508 = 2 \times 1254 + 0$

Remainder is $0$, so $2508$ is an even number.

Answer: Even (remainder $0$ when divided by $2$)

Multiples of $19$: $19 \times 1, 19 \times 2, 19 \times 3, \ldots$

$19 \times 7 = 133$ is less than $150$

$19 \times 8 = 152$, $19 \times 9 = 171$, $19 \times 10 = 190$

$19 \times 11 = 209$, $19 \times 12 = 228$, $19 \times 13 = 247$

$19 \times 14 = 266$ is more than $250$

Multiples of $19$ between $150$ and $250$: $152, 171, 190, 209, 228, 247$

Answer: $152, 171, 190, 209, 228, 247$

$35 = 1 \times 35$

$= 5 \times 7$

$\therefore$ Factors of $35$ are $1, 5, 7, 35$

Answer: $1, 5, 7, 35$

$126 = 1 \times 126$

$= 2 \times 63$

$= 3 \times 42$

$= 6 \times 21$

$= 7 \times 18$

$= 9 \times 14$

$4, 5, 8, 10, 11$ do not divide $126$ (they leave a remainder).

$\therefore$ Divisors of $126$ are $1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126$

Number of divisors $= 12$

Answer: $1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126$ ($12$ divisors)

A symbol which has a fixed value is called a constant.

Examples: $5$, $8$

Answer: A symbol which has a fixed value; e.g. $5, 8$

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

$-6 \times c = -6c$

Answer: $-6c$

(i) $2t + 6t + 9t = (2 + 6 + 9)t = 17t$

(ii) $10t - 3t = (10 - 3)t = 7t$

Answer: (i) $17t$ (ii) $7t$

(i) $z + z + z + z + z$ is $z$ added $5$ times: $z + z + z + z + z = 5z$

(ii) $z \times z \times z \times z \times z$ is $z$ multiplied $5$ times: $z \times z \times z \times z \times z = z^{5}$

Answer: (i) $5z$ (ii) $z^{5}$

$53^{2} = 53 \times 53$

$53 \times 3 = 159$

$53 \times 5 = 265$, written one place to the left: $2650$

$159 + 2650 = 2809$

$\therefore 53^{2} = 2809$

Answer: $2809$

$12^{3} = 12 \times 12 \times 12$

$12 \times 12 = 144$, $144 \times 12 = 1728$

$\therefore 12^{3} = 1728$

Answer: $1728$

(i) Divide by $7$ again and again:

$2401 \div 7 = 343$, $343 \div 7 = 49$, $49 \div 7 = 7$

$2401 = 7 \times 7 \times 7 \times 7 = 7^{4}$

(ii) Divide by $49$ again and again:

$2401 \div 49 = 49$

$2401 = 49 \times 49 = 49^{2}$

Answer: (i) $7^{4}$ (ii) $49^{2}$

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 $40759$ is $9$, so $40759$ is not divisible by $2$.

Answer: No, the unit digit is $9$

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

Sum of the digits of $88571$: $8 + 8 + 5 + 7 + 1 = 29$

$29$ is not divisible by $3$ ($29 = 3 \times 9 + 2$), so $88571$ is not divisible by $3$.

Answer: No, the sum of the digits is $29$

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

$1000$: $1 + 0 + 0 + 0 = 1$, not divisible by $3$

$1001$: $1 + 0 + 0 + 1 = 2$, not divisible by $3$

$1002$: $1 + 0 + 0 + 2 = 3$, divisible by $3$

$\therefore$ The smallest $4$-digit number divisible by $3$ is $1002$.

Answer: $1002$

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

Last two digits of $46290$ together: $90$; $90 = 4 \times 22 + 2$, so $46290$ is not divisible by $4$.

Answer: No, the last two digits $90$ 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 $25051$ is $1$, so $25051$ is not divisible by $5$.

Answer: No, the unit digit is $1$

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

The digit at the unit place of $50412$ is $2$, so $50412$ is divisible by $2$.

Sum of the digits of $50412$: $5 + 0 + 4 + 1 + 2 = 12$

$12$ is divisible by $3$ ($12 = 3 \times 4$), so $50412$ is divisible by $3$.

$\therefore$ $50412$ 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 $0$).

$499$: $49 - 9(2) = 49 - 18 = 31$; $31 = 7 \times 4 + 3$, not a multiple of $7$, so $499$ is not divisible by $7$.

$252$: $25 - 2(2) = 25 - 4 = 21$; $21 = 7 \times 3$, a multiple of $7$, so $252$ is divisible by $7$.

$663$: $66 - 3(2) = 66 - 6 = 60$; $60 = 7 \times 8 + 4$, not a multiple of $7$, so $663$ is not divisible by $7$.

Answer: $252$ is divisible by $7$; $499$ and $663$ are not.

By $2$: The digit at the unit place of $70328$ is $8$, so $70328$ is divisible by $2$.

By $3$: Sum of the digits of $70328$: $7 + 0 + 3 + 2 + 8 = 20$; $20$ is not divisible by $3$ ($20 = 3 \times 6 + 2$), so $70328$ is not divisible by $3$.

By $4$: Last two digits of $70328$ together: $28$; $28 = 4 \times 7$, so $70328$ is divisible by $4$.

By $5$: The digit at the unit place of $70328$ is $8$, so $70328$ is not divisible by $5$.

By $6$: $70328$ is not divisible by both $2$ and $3$, so it is not divisible by $6$.

$\therefore$ $70328$ is divisible by $2$ and $4$.

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