Practice Questions for Algebra - Number Theory

1. PE 2 - Numbers | Algebra - Number Theory

How many even numbers from 10 to 10,000 are not perfect squares (including both)?

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2. PE 2 - Numbers | Algebra - Number Theory

If the last 2 digits of a four-digit number are interchanged, the new number obtained is greater than the original number by 72. What is the difference between the last two digits of the number?

(a)
9
(b)
1
(c)
6
(d)
8
(e)
Cannot be determined

3. PE 2 - Numbers | Algebra - Number Theory

LCM of two numbers x and y is 143. Find the value of |x – y|, given that y > x, x > 1.

(a)
30
(b)
15
(c)
41
(d)
165
(e)
Cannot be determined

4. PE 2 - Numbers | Algebra - Number Theory

XX and YY are 2-digits numbers such that their sum is ABA. Find the value of A + B + A?

(a)
4
(b)
3
(c)
2
(d)
1
(e)
5

5. PE 2 - Numbers | Algebra - Number Theory

Three consecutive whole numbers are such that product of smallest and the largest number is one less than the square of the middle number. Find the middle number.

(a)
6
(b)
18
(c)
12
(d)
31
(e)
All of these

6. PE 2 - Numbers | Algebra - Number Theory

A = 1¹ × 2² × 3³ × 4⁴ × … 100¹⁰⁰. How many zeroes will be there at the end of A?

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7. PE 2 - Numbers | Algebra - Number Theory

If x is a prime such that (x⁴ + 1) is also a prime, then how many values are possible for x.

(a)
0
(b)
1
(c)
2
(d)
more than 2 values

8. PE 2 - Numbers | Algebra - Number Theory

In how many ways can 14 be written as a product of two or more distinct integers, without considering two integers with the same absolute value but opposite signs?

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9. PE 2 - Numbers | Algebra - Number Theory

‘N’ is a two-digit number less than 50 such that when the unit’s digit of ‘N’ is erased, the resulting number is a factor of ‘N’. How many possible values of ‘N’ are there?

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Answer: 22

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Explanation :

Let N = ‘ab’, where a and b are digits
N = 10a + b

Now, since a is a factor of ‘ab’, a should divide (10a + b) completely.
i.e., $R [\frac{10 a + b}{a}]$ = 0

⇒ $R [\frac{10 a}{a}]$ + $R [\frac{b}{a}]$ = 0

⇒ $R [\frac{b}{a}]$ = 0

∴ ‘b’ should be divisible by a.

If a = 1, b can be 0, 1, 2, 3, …, 9 ⇒ 10 possible values
If a = 2, b can be 0, 2, 4, 6, 8 ⇒ 5 possible values
If a = 3, b can be 0, 3, 6, 9 ⇒ 4 possible values
If a = 4, b can be 0, 4, 8 ⇒ 3 possible values

Number of possible values of ‘N’ = 10 + 5 + 4 + 3 = 22.

Hence, 22.

10. PE 2 - Numbers | Algebra - Number Theory

How many integers exist such that not only are they multiples of A = 2020²⁰²⁰ but also are factors of B = 2020²⁰²²?

(a)
36
(b)
48
(c)
45
(d)
None of these

PE 2 - Numbers | Algebra - Number Theory

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