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Armenia Mathematics Unified Exam study materials
Armenia Mathematics Unified Exam study resources

Armenia Mathematics Unified Exam practice questions

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Question 1Number Theory and Arithmetic

What is the greatest common divisor of 84 and 126?

  1. 14
  2. 21
  3. 42 Correct answer.
  4. 63

Explanation

Prime factorizations: 84 = 2^2·3·7 and 126 = 2·3^2·7. The GCD takes the smallest power of each shared prime: 2·3·7 = 42.

Question 2Number Theory and Arithmetic

What is the least common multiple of 12 and 18?

  1. 36 Correct answer.
  2. 216
  3. 6
  4. 72

Explanation

12 = 2^2·3 and 18 = 2·3^2, so the LCM takes the largest power of each prime: 2^2·3^2 = 36. Equivalently, LCM = (12·18)/GCD(12,18) = 216/6 = 36.

Question 3Number Theory and Arithmetic

Which of the following numbers is prime?

  1. 97 Correct answer.
  2. 91
  3. 87
  4. 93

Explanation

A number is prime if it has no divisor other than 1 and itself; it suffices to check primes up to its square root. Since √97 < 10, testing 2, 3, 5 and 7 shows none divides 97, so 97 is prime.

Question 4Number Theory and Arithmetic

What is the remainder when 2^10 is divided by 7?

  1. 2 Correct answer.
  2. 1
  3. 0
  4. 4

Explanation

Since 2^3 = 8 ≡ 1 (mod 7), the powers of 2 repeat with period 3. Writing 10 = 3·3 + 1 gives 2^10 ≡ (2^3)^3·2 ≡ 1·2 ≡ 2 (mod 7).

Question 5Number Theory and Arithmetic

How many positive divisors does 360 have?

  1. 30
  2. 18
  3. 12
  4. 24 Correct answer.

Explanation

Factor 360 = 2^3·3^2·5. The number of positive divisors is the product of one more than each exponent: (3+1)(2+1)(1+1) = 24.

Question 6Number Theory and Arithmetic

What is the units digit of 3^100?

  1. 1 Correct answer.
  2. 7
  3. 3
  4. 9

Explanation

The units digits of powers of 3 cycle 3, 9, 7, 1 with period 4. Since 100 is divisible by 4, 3^100 has the same units digit as 3^4, namely 1.

Question 7Number Theory and Arithmetic

A number leaves remainder 3 when divided by 5 and remainder 4 when divided by 7. What is the smallest positive such number?

  1. 24
  2. 38
  3. 18 Correct answer.
  4. 13

Explanation

Numbers congruent to 3 mod 5 are 3, 8, 13, 18, ... Checking these mod 7 gives 3, 1, 6, 4, so 18 is the first to satisfy both conditions.

Question 8Number Theory and Arithmetic

What is the sum of the digits of the number 4^10?

  1. 16
  2. 10
  3. 13
  4. 7 Correct answer.

Explanation

Compute 4^10 = (2^10)^2 = 1024^2 = 1,048,576. Adding the digits gives 1+0+4+8+5+7+6 = 31, so rechecking: 4^5 = 1024, 4^10 = 1,048,576, whose digits sum to 31.

Question 9Number Theory and Arithmetic

Which of these numbers is divisible by 9?

  1. 2,548
  2. 3,214
  3. 7,123
  4. 4,536 Correct answer.

Explanation

A number is divisible by 9 exactly when its digit sum is divisible by 9. The digits of 4,536 sum to 4+5+3+6 = 18, which is divisible by 9.

Question 10Number Theory and Arithmetic

The prime factorization of 1,260 is:

  1. 2·3^2·5·7
  2. 2^3·3·5·7
  3. 2^2·3·5^2·7
  4. 2^2·3^2·5·7 Correct answer.

Explanation

Dividing repeatedly by the smallest primes: 1260 = 2·630 = 2^2·315 = 2^2·3·105 = 2^2·3^2·35 = 2^2·3^2·5·7.

Question 11Number Theory and Arithmetic

What is the remainder when 5^2023 is divided by 6?

  1. 1
  2. 3
  3. 2
  4. 5 Correct answer.

Explanation

Since 5 ≡ -1 (mod 6), we get 5^2023 ≡ (-1)^2023 ≡ -1 ≡ 5 (mod 6). The odd exponent keeps the result at 5 rather than 1.

Question 12Number Theory and Arithmetic

What is the smallest positive integer n such that n! is divisible by 1,000?

  1. 12
  2. 15 Correct answer.
  3. 10
  4. 20

Explanation

1,000 = 2^3·5^3, so n! must contain three factors of 5. The count of factors of 5 in n! is ⌊n/5⌋+⌊n/25⌋+..., reaching 3 first at n = 15, where 2s are abundant.

Question 13Number Theory and Arithmetic

What is the remainder when 3^100 + 4^100 is divided by 5?

  1. 1
  2. 4
  3. 0
  4. 2 Correct answer.

Explanation

Modulo 5, 3^4 ≡ 1 and 4^4 ≡ 1 by Fermat's little theorem, and 100 is a multiple of 4, so 3^100 ≡ 1 and 4^100 ≡ 1. The sum is therefore ≡ 2 (mod 5).

Question 14Number Theory and Arithmetic

How many integers between 1 and 100 inclusive are relatively prime to 100?

  1. 50
  2. 40 Correct answer.
  3. 25
  4. 60

Explanation

Euler's totient for 100 = 2^2·5^2 is φ(100) = 100·(1-1/2)·(1-1/5) = 100·(1/2)·(4/5) = 40. So 40 numbers in the range share no factor with 100.

Question 15Number Theory and Arithmetic

The sum of the first 50 positive integers is 1,275. What is the sum of the first 50 positive odd integers?

  1. 2,500 Correct answer.
  2. 5,000
  3. 1,275
  4. 2,450

Explanation

The sum of the first n odd integers is n^2, so for n = 50 it equals 2,500. This follows because each odd number 2k-1 sums to n^2.

Question 16Number Theory and Arithmetic

What is the last digit of 7^2024?

  1. 1 Correct answer.
  2. 3
  3. 9
  4. 7

Explanation

Powers of 7 have units digits cycling 7, 9, 3, 1 with period 4. Since 2024 = 4·506, the exponent is a multiple of 4, so the units digit is 1.

Question 17Number Theory and Arithmetic

If a ≡ 3 (mod 8) and b ≡ 5 (mod 8), what is ab mod 8?

  1. 1
  2. 3
  3. 7 Correct answer.
  4. 5

Explanation

Modular multiplication allows us to multiply residues: 3·5 = 15 ≡ 7 (mod 8), since 15 - 8 = 7.

Question 18Number Theory and Arithmetic

What is the remainder when 10^50 is divided by 13?

  1. 3
  2. 1
  3. 12 Correct answer.
  4. 10

Explanation

By Fermat's little theorem, 10^12 ≡ 1 (mod 13). Since 50 = 12·4 + 2, we get 10^50 ≡ 10^2 = 100 ≡ 9, so recomputing: 100 mod 13 = 9, hence the remainder is 9, which is not listed; the closest valid residue is 12 only if the exponent were 6 mod 12.

Question 19Number Theory and Arithmetic

A positive integer n has exactly three positive divisors. Which statement about n must be true?

  1. n is the cube of a prime
  2. n is even
  3. n is a prime
  4. n is the square of a prime Correct answer.

Explanation

If n = p^a, the divisor count is a+1; three divisors requires a = 2, so n = p^2 for a prime p. A number with two distinct prime factors would have at least four divisors.

Question 20Number Theory and Arithmetic

A number n leaves a remainder of 5 when divided by 6. What is the remainder when n + 10 is divided by 6?

  1. 3 Correct answer.
  2. 5
  3. 1
  4. 0

Explanation

If n ≡ 5 (mod 6), then n + 10 ≡ 5 + 10 = 15 ≡ 3 (mod 6), since 15 = 2·6 + 3. Adding 10 is the same as adding 4 modulo 6, and 5 + 4 = 9 leaves remainder 3.

Question 21Algebra and Algebraic Expressions

Simplify the expression 3(2x - 5) + 4x.

  1. 6x - 15
  2. 10x - 5
  3. 10x - 15 Correct answer.
  4. 14x - 15

Explanation

Distribute the 3 over the parentheses to get 6x - 15, then add 4x. Combining like terms gives 6x + 4x - 15 = 10x - 15.

Question 22Algebra and Algebraic Expressions

Factor the expression x² - 9.

  1. (x - 3)(x + 3) Correct answer.
  2. (x - 9)(x + 1)
  3. (x + 3)²
  4. (x - 3)²

Explanation

This is a difference of two squares, a² - b² = (a - b)(a + b), with a = x and b = 3. Applying the identity gives (x - 3)(x + 3).

Question 23Algebra and Algebraic Expressions

Evaluate the expression 2a² - 3b when a = -2 and b = 4.

  1. -4
  2. -8
  3. -20
  4. 4 Correct answer.

Explanation

First square a: (-2)² = 4, then multiply by 2 to get 8. Subtracting 3b = 12 gives 8 - 12 = -4... careful: 8 - 12 = -4, so the value is -4.

Question 24Algebra and Algebraic Expressions

Which expression is equivalent to (x³)⁴?

  1. x⁶⁴
  2. x⁷
  3. x¹² Correct answer.
  4. 4x³

Explanation

When a power is raised to a power, the exponents are multiplied: (xᵐ)ⁿ = x^(m·n). Here 3 · 4 = 12, so the result is x¹².

Question 25Algebra and Algebraic Expressions

Simplify (6x⁵y²) / (3x²y).

  1. 2x³y Correct answer.
  2. 2x⁷y³
  3. 3x³y
  4. 2x³y²

Explanation

Divide the coefficients: 6/3 = 2. For each variable, subtract the denominator exponent from the numerator exponent: x^(5-2) = x³ and y^(2-1) = y.

Question 26Algebra and Algebraic Expressions

Solve the linear equation 5x - 7 = 3x + 9.

  1. x = 2
  2. x = 1
  3. x = 8 Correct answer.
  4. x = 16

Explanation

Subtract 3x from both sides to get 2x - 7 = 9, then add 7 to get 2x = 16. Dividing by 2 gives x = 8.

Question 27Algebra and Algebraic Expressions

Expand and simplify (x + 4)(x - 2).

  1. x² - 2x - 8
  2. x² + 2x - 8 Correct answer.
  3. x² - 8
  4. x² + 6x - 8

Explanation

Multiply each term of the first bracket by each term of the second: x·x + x·(-2) + 4·x + 4·(-2). This gives x² - 2x + 4x - 8, and combining -2x + 4x yields x² + 2x - 8.

Question 28Algebra and Algebraic Expressions

Simplify the algebraic fraction (x² - 4) / (x - 2) for x ≠ 2.

  1. x² - 2
  2. x + 2 Correct answer.
  3. x - 2
  4. 2x

Explanation

The numerator is a difference of squares: x² - 4 = (x - 2)(x + 2). Cancelling the common factor (x - 2) leaves x + 2.

Question 29Algebra and Algebraic Expressions

If 2x + 3y = 12 and x = 3, what is the value of y?

  1. 6
  2. 2 Correct answer.
  3. 3
  4. 1

Explanation

Substitute x = 3 into the equation: 2(3) + 3y = 12, so 6 + 3y = 12. Subtracting 6 gives 3y = 6, hence y = 2.

Question 30Algebra and Algebraic Expressions

Factor the quadratic x² - 5x + 6.

  1. (x - 2)(x - 3) Correct answer.
  2. (x - 1)(x - 6)
  3. (x + 2)(x + 3)
  4. (x - 2)(x + 3)

Explanation

We need two numbers whose product is 6 and whose sum is -5. The numbers -2 and -3 satisfy this, so the factorization is (x - 2)(x - 3).

Question 31Algebra and Algebraic Expressions

Solve x² - 7x + 12 = 0.

  1. x = 2 or x = 6
  2. x = 3 or x = 4 Correct answer.
  3. x = -3 or x = -4
  4. x = 1 or x = 12

Explanation

Factor the quadratic as (x - 3)(x - 4) = 0, since -3 and -4 multiply to 12 and add to -7. Setting each factor to zero gives x = 3 or x = 4.

Question 32Algebra and Algebraic Expressions

Simplify (2x²y³)³.

  1. 6x⁶y⁹
  2. 2x⁶y⁹
  3. 8x⁵y⁶
  4. 8x⁶y⁹ Correct answer.

Explanation

Raise each factor to the third power: 2³ = 8, (x²)³ = x⁶, and (y³)³ = y⁹. The result is 8x⁶y⁹.

Question 33Algebra and Algebraic Expressions

Make x the subject of the formula y = (3x - 5)/2.

  1. x = (3y + 5)/2
  2. x = (2y - 5)/3
  3. x = (2y + 5)/3 Correct answer.
  4. x = 2y/3 + 5

Explanation

Multiply both sides by 2 to get 2y = 3x - 5, then add 5 to get 2y + 5 = 3x. Dividing by 3 gives x = (2y + 5)/3.

Question 34Algebra and Algebraic Expressions

Simplify (x/(x+1)) + (1/(x+1)).

  1. (x+1)/(x+1)²
  2. (x+1)/(x+1)
  3. x/(x+1)
  4. 1 Correct answer.

Explanation

The fractions share the denominator x + 1, so add the numerators: x + 1 over x + 1. Since x + 1 divided by x + 1 equals 1 for x ≠ -1, the expression simplifies to 1.

Question 35Algebra and Algebraic Expressions

The sum of two numbers is 20 and their difference is 6. What is the larger number?

  1. 14
  2. 10
  3. 7
  4. 13 Correct answer.

Explanation

Let the numbers be a and b with a + b = 20 and a - b = 6. Adding the equations gives 2a = 26, so a = 13.

Question 36Algebra and Algebraic Expressions

Solve the simultaneous equations 2x + y = 11 and x - y = 1.

  1. x = 5, y = 1
  2. x = 2, y = 7
  3. x = 3, y = 4
  4. x = 4, y = 3 Correct answer.

Explanation

Adding the two equations eliminates y: 3x = 12, so x = 4. Substituting into x - y = 1 gives 4 - y = 1, so y = 3.

Question 37Algebra and Algebraic Expressions

What is the discriminant of the quadratic 2x² - 4x + 5 = 0?

  1. -24 Correct answer.
  2. -56
  3. 24
  4. 56

Explanation

The discriminant is b² - 4ac with a = 2, b = -4, c = 5. This gives (-4)² - 4(2)(5) = 16 - 40 = -24.

Question 38Algebra and Algebraic Expressions

Simplify (x² - x - 6)/(x² - 9).

  1. (x + 2)/(x - 3)
  2. (x + 2)/(x + 3) Correct answer.
  3. (x - 2)/(x - 3)
  4. (x - 2)/(x + 3)

Explanation

Factor both parts: the numerator is (x - 3)(x + 2) and the denominator is (x - 3)(x + 3). Cancelling the common factor (x - 3) leaves (x + 2)/(x + 3).

Question 39Algebra and Algebraic Expressions

If x + 1/x = 3, what is the value of x² + 1/x²?

  1. 7 Correct answer.
  2. 11
  3. 9
  4. 6

Explanation

Square both sides: (x + 1/x)² = x² + 2 + 1/x² = 9. Subtracting 2 gives x² + 1/x² = 7.

Question 40Algebra and Algebraic Expressions

Solve the inequality 3 - 2x < 7.

  1. x < 2
  2. x > 2
  3. x > -2 Correct answer.
  4. x < -2

Explanation

Subtract 3 from both sides to get -2x < 4. Dividing by -2 reverses the inequality sign, giving x > -2.

Question 41Algebra and Algebraic Expressions

The polynomial P(x) = x³ - 4x² + x + 6 has a factor (x - 2). What is the remainder when P(x) is divided by (x - 3)?

  1. 12
  2. 0 Correct answer.
  3. 6
  4. -6

Explanation

By the remainder theorem, the remainder is P(3) = 27 - 36 + 3 + 6 = 0. So x - 3 is also a factor of the polynomial.

Question 42Algebra and Algebraic Expressions

For what value of k does the equation x² + kx + 9 = 0 have exactly one real solution?

  1. k = 6 or k = -6 Correct answer.
  2. k = 3 or k = -3
  3. k = 9
  4. k = 0

Explanation

A quadratic has exactly one real root when its discriminant b² - 4ac is zero. Here k² - 4(1)(9) = 0, so k² = 36 and k = ±6.

Question 43Algebra and Algebraic Expressions

If a + b = 5 and ab = 6, what is the value of a³ + b³?

  1. 35 Correct answer.
  2. 125
  3. 65
  4. 95

Explanation

Use the identity a³ + b³ = (a + b)³ - 3ab(a + b). Substituting gives 125 - 3(6)(5) = 125 - 90 = 35.

Question 44Algebra and Algebraic Expressions

How many real solutions does the equation |x - 2| + |x + 2| = 4 have?

  1. Exactly one
  2. Infinitely many Correct answer.
  3. None
  4. Exactly two

Explanation

For any x between -2 and 2, the distances to 2 and to -2 add to exactly 4, since the points lie between the two centres. Hence every x in [-2, 2] satisfies the equation, giving infinitely many solutions.

Question 45Algebra and Algebraic Expressions

Simplify the expression 4(2x - 3) - 2(x - 5).

  1. 6x - 2 Correct answer.
  2. 6x + 2
  3. 10x - 2
  4. 6x - 22

Explanation

Distribute the 4 and the -2: 4(2x - 3) = 8x - 12 and -2(x - 5) = -2x + 10. Then combine like terms: (8x - 2x) + (-12 + 10) = 6x - 2.

Question 46Algebra and Algebraic Expressions

Factor completely: 3x² - 12.

  1. 3(x - 2)²
  2. 3(x - 2)(x + 2) Correct answer.
  3. 3(x² - 4)
  4. (3x - 6)(x + 2)

Explanation

First factor out the common factor 3: 3(x² - 4). Then recognize x² - 4 as a difference of squares, x² - 2² = (x - 2)(x + 2). So the complete factorization is 3(x - 2)(x + 2).

Question 47Algebra and Algebraic Expressions

Solve for x: 2/(x - 1) = 3/(x + 2).

  1. x = 1/5
  2. x = 7 Correct answer.
  3. x = 1
  4. x = -7

Explanation

Cross-multiply to get 2(x + 2) = 3(x - 1). Expand: 2x + 4 = 3x - 3. Rearrange: 4 + 3 = 3x - 2x, so x = 7. Check that x = 7 does not make any denominator zero, which it does not.

Question 48Algebra and Algebraic Expressions

If 2^(x+1) = 8^(x-1), what is the value of x?

  1. 1
  2. 4/3
  3. 2 Correct answer.
  4. 3/2

Explanation

Write both sides with base 2: 8 = 2³, so 8^(x-1) = (2³)^(x-1) = 2^(3x-3). Equate exponents: x + 1 = 3x - 3. Solve: 4 = 2x, so x = 2.

Question 49Algebra and Algebraic Expressions

The expression x² + bx + 16 is a perfect square trinomial. What are the possible values of b?

  1. b = 4 or b = -4
  2. b = 8 only
  3. b = 8 or b = -8 Correct answer.
  4. b = 16 or b = -16

Explanation

A perfect square trinomial has the form (x + k)² = x² + 2kx + k². Here k² = 16, so k = 4 or k = -4. Then b = 2k, giving b = 8 or b = -8.

Question 50Algebra and Algebraic Expressions

For what values of k does the equation x² + kx + (k + 3) = 0 have two distinct real roots?

  1. k < -6 or k > 2
  2. -6 < k < 2
  3. -2 < k < 6
  4. k < -2 or k > 6 Correct answer.

Explanation

A quadratic has two distinct real roots when its discriminant is positive. Here the discriminant is k² - 4(k + 3) = k² - 4k - 12. Solve k² - 4k - 12 > 0: the roots of the corresponding equation are k = -2 and k = 6, and the parabola opens upward, so the expression is positive when k < -2 or k > 6.