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PE 1 - Numbers | Algebra - Number Theory

1. PE 1 - Numbers | Algebra - Number Theory

The least number which on division by 25 leaves the remainder 15, on division by 35 leaves the remainder 25 and on division by 45 leaves the remainder 35 is

(a)
1565
(b)
1575
(c)
1585
(d)
1595
(e)
None of these

2. PE 1 - Numbers | Algebra - Number Theory

A group of students is divided into groups of 3, 4 and 5, and each time one student is left out. The least number of students in the heap is

(a)
31
(b)
41
(c)
51
(d)
61
(e)
71

3. PE 1 - Numbers | Algebra - Number Theory

What is the highest power of 288 that divides 35!?

(a)
2
(b)
6
(c)
4
(d)
5
(e)
None of these

Answer: Option B

Explanation :

288 = 2⁵ × 3²

The highest power of 3 in 35! is $R [\frac{35}{3}]$ + $R [\frac{35}{3^{2}}]$ + $R [\frac{35}{3^{3}}]$ = 11 + 3 + 1 = 15.

∴ The highest power of 3² will be $\frac{15}{2}$ = 7.

The highest power of 2 in 35! is $R [\frac{35}{2}]$ + $R [\frac{35}{2^{2}}]$ + $R [\frac{35}{2^{3}}]$ + $R [\frac{35}{2^{4}}]$ = 17 + 8 + 4 + 2 + 1 = 32.

∴ The highest power of 2⁵ will be $\frac{32}{5}$ = 6.

⇒ The highest power of 288 will be lower of 7 and 6 i.e., 6.

Hence, option (b).

4. PE 1 - Numbers | Algebra - Number Theory

369 has been divided into three parts such that half of the first part, one-third of the second part and one-fourth of the third part are equal. The largest part is

Backspace

7 8 9 4 5 6 1 2 3 0 . -

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Answer: 164

Explanation :

$\frac{A}{2}$ = $\frac{B}{3}$ = $\frac{C}{4}$ = x

⇒ A = 2x; B = 3x; C = 4x

⇒ A : B : C = 2 : 3 : 4

Largest part = 369 × $\frac{4}{9}$ = 164

Hence, 164.

5. PE 1 - Numbers | Algebra - Number Theory

Seven bells commence tolling together and toll at intervals of 3, 6, 9, 12, 15, 18 and 21 s respectively. In 300 mins, how many times do they toll together?

(a)
4
(b)
11
(c)
10
(d)
15
(e)
None of these

6. PE 1 - Numbers | Algebra - Number Theory

When ‘n’ is divided by 5 the remainder is 3. What is the remainder when n² is divided by 5?

(a)
2
(b)
1
(c)
3
(d)
4
(e)
0

Answer: Option D

Explanation :

n = 5k + 3

⇒ n² = 25k² + 30k + 9

= $R [\frac{25 k^{2} + 30 k + 9}{5}]$

= $R [\frac{25 k^{2}}{5}]$ + $R [\frac{30 k}{5}]$ + $R [\frac{9}{5}]$ = 0 + 0 + 4 = 4.

Hence, option (d).

7. PE 1 - Numbers | Algebra - Number Theory

The expression $\frac{1}{1 \times 2}$ + $\frac{1}{2 \times 3}$ + $\frac{1}{3 \times 4}$ + ... + $\frac{1}{n \times (n + 1)}$ for any natural number n, is

(a)
always less than 1
(b)
always greater than 2
(c)
always equal to 1
(d)
always lies between 1 and 2
(e)
None of these

Answer: Option A

Explanation :

N = $\frac{1}{1 \times 2}$ + $\frac{1}{2 \times 3}$ + $\frac{1}{3 \times 4}$ + ... + $\frac{1}{n \times (n + 1)}$

N = $\frac{2 - 1}{1 \times 2}$ + $\frac{3 - 2}{2 \times 3}$ + $\frac{4 - 3}{3 \times 4}$ + ... + $\frac{(n + 1) - n}{n \times (n + 1)}$

N = $\frac{2}{1 \times 2}$ - $\frac{1}{1 \times 2}$ + $\frac{3}{2 \times 3}$ - $\frac{2}{2 \times 3}$ + $\frac{4}{3 \times 4}$ - $\frac{3}{3 \times 4}$ + ... + $\frac{(n + 1)}{n \times (n + 1)}$ - $\frac{n}{n \times (n + 1)}$

N = $\frac{1}{1}$ - $\frac{1}{2}$ + $\frac{1}{2}$ - $\frac{1}{3}$ + $\frac{1}{3}$ - $\frac{1}{4}$ + ... + $\frac{1}{n}$ - $\frac{1}{(n + 1)}$

N = $\frac{1}{1}$ - $\frac{1}{n}$

Since 'n' is a natural numbers N will always be less than 1.

Hence, option (a).

8. PE 1 - Numbers | Algebra - Number Theory

The ratio between a two-digit number and the sum of the digits of that number is 4 : 1. If the digit in the unit’s place is twice the digit in the ten’s place, what is that number?

(a)
48
(b)
63
(c)
84
(d)
Cannot be determined

Answer: Option D

Explanation :

Let tens digit be x. Then, units digit = 2x

Sum of the digits = x + 2x = 3x

Number = 10x + 2x = 12x

Now, $\frac{12 x}{3 x}$ = $\frac{4}{1}$

This is true for all values of x.

Hence, option (d).

9. PE 1 - Numbers | Algebra - Number Theory

What is the greatest positive power of 5 that divides 40! exactly?

(a)
5
(b)
6
(c)
7
(d)
8
(e)
9

Answer: Option E

Explanation :

The number of 5s in 30! is calculated as follows:

$R [\frac{40}{5}]$ + $R [\frac{40}{5^{2}}]$ = 8 + 1 = 9

Hence, option (e).

10. PE 1 - Numbers | Algebra - Number Theory

If the sum of two natural numbers is multiplied by each number separately, the products so obtained are 2772 and 1584. What is the difference between the numbers?

(a)
18
(b)
22
(c)
26
(d)
35
(e)
27