[ LET 46 ] Board Licensure Examination for Professional Teachers REVIEWER - Mathematics

Professional Education.

Instructions:

Please answer each question to the best of your ability. Each question is multiple choice, and only one answer per question is correct. Select the most appropriate answer from the options provided. There are 50 questions in total.

When you have completed all questions, click the "Submit" button at the bottom of the page to see your score. Good luck!

1. What is the largest prime factor of 1001?

A) 11
B) 7
C) 13
D) 17

2. How many distinct prime factors does 360 have?

A) 3
B) 2
C) 4
D) 5

3. What is the sum of the digits of 123456789?

A) 36
B) 45
C) 37
D) 40

4. Find the value of \( x \) in the congruence \( 7x \equiv 1 \mod 26 \).

A) 15
B) 4
C) 5
D) 18

5. What is the smallest positive integer that is both a multiple of 6 and a multiple of 8?

A) 24
B) 48
C) 36
D) 18

6. What is the greatest common divisor (gcd) of 252 and 105?

A) 21
B) 21
C) 7
D) 14

7. What is the remainder when \( 2^{100} \) is divided by 11?

A) 1
B) 10
C) 9
D) 8

8. What is the largest perfect square that divides 120?

A) 36
B) 16
C) 4
D) 9

9. If \( p \) is a prime number, what is the value of \( p^2 - 1 \)?

A) Even
B) Odd
C) Prime
D) Composite

10. What is the value of \( 3^4 - 4^3 \)?

A) -7
B) 0
C) 7
D) 16

11. What is the sum of the first 20 prime numbers?

A) 77
B) 85
C) 98
D) 112

12. If \( n \) is an integer, for how many values of \( n \) is \( n^2 - 4n + 3 = 0 \) true?

A) 2
B) 3
C) 1
D) 0

13. What is the number of divisors of 48?

A) 8
B) 12
C) 10
D) 9

14. How many ways can you express 12 as a sum of two positive integers?

A) 11
B) 10
C) 12
D) 9

15. What is the least common multiple (LCM) of 4 and 6?

A) 10
B) 12
C) 18
D) 24

16. What is the number of integers \( n \) such that \( 1 \leq n \leq 100 \) and \( n^2 \equiv 1 \mod 5 \)?

A) 20
B) 40
C) 50
D) 60

17. What is the highest power of 2 that divides 48?

A) 16
B) 4
C) 2
D) 8

18. How many integers \( n \) satisfy \( n^2 - 5n + 6 = 0 \)?

A) 0
B) 2
C) 1
D) 3

19. What is the sum of the first \( n \) odd integers?

A) \( n^2 \)
B) \( 2n - 1 \)
C) \( n(n + 1) \)
D) \( n^2 + n \)

20. What is the smallest prime greater than 100?

A) 101
B) 103
C) 107
D) 109

21. What is the number of positive divisors of 60?

A) 8
B) 10
C) 12
D) 14

22. What is the largest integer \( n \) such that \( n^2 < 50 \)?

A) 7
B) 6
C) 8
D) 9

23. Find \( x \) such that \( 3x \equiv 2 \mod 5 \).

A) 4
B) 1
C) 3
D) 2

24. What is the value of \( \phi(36) \)?

A) 12
B) 24
C) 18
D) 9

25. What is the next prime number after 19?

A) 23
B) 29
C) 31
D) 37

26. What is the least number \( n \) such that \( n^3 \equiv 1 \mod 4 \)?

A) 3
B) 1
C) 0
D) 2

27. If \( p \) is a prime, what is \( p^2 + p + 1 \) always?

A) Even
B) Odd
C) Composite
D) Prime

28. What is the solution to \( x^2 \equiv 1 \mod 12 \)?

A) 1, 5
B) 1, 11
C) 5, 7
D) 0, 6

29. What is the last digit of \( 3^{100} \)?

A) 3
B) 1
C) 9
D) 7

30. How many perfect squares are less than 1000?

A) 30
B) 31
C) 31
D) 32

31. What is the sum of the digits of 12345?

A) 10
B) 12
C) 15
D) 20

32. How many integers \( n \) satisfy \( n^2 + n + 1 = 0 \) in \( \mathbb{Z} \)?

A) 1
B) 0
C) 2
D) 3

33. What is \( 10! \) mod 7?

A) 3
B) 1
C) 0
D) 5

34. If \( n \equiv 3 \mod 5 \) and \( n \equiv 4 \mod 7 \), what is \( n \mod 35 \)?

A) 10
B) 24
C) 17
D) 18

35. What is the smallest integer \( x \) such that \( x^2 \equiv 2 \mod 11 \)?

A) 5
B) 3
C) 1
D) 4

36. What is the largest prime divisor of 84?

A) 7
B) 5
C) 2
D) 3

37. Find the number of integers \( n \) such that \( n^2 - n - 6 = 0 \).

A) 2
B) 3
C) 4
D) 5

38. What is \( \gcd(60, 48) \)?

A) 6
B) 12
C) 4
D) 8

39. What is the remainder when \( 25^3 \) is divided by 8?

A) 1
B) 3
C) 5
D) 7

40. If \( n \) is a positive integer, what is the largest value of \( k \) such that \( 2^k \) divides \( n! \) for \( n = 10 \)?

A) 5
B) 8
C) 10
D) 7

41. What is the smallest integer \( n \) such that \( n! \) ends with three zeros?

A) 10
B) 15
C) 20
D) 25

42. What is the least positive integer \( n \) such that \( n^3 \equiv 1 \mod 10 \)?

A) 5
B) 3
C) 1
D) 9

43. What is the sum of the first \( n \) even integers?

A) \( n(n + 1) \)
B) \( n(n + 1) \)
C) \( n^2 \)
D) \( 2^n \)

44. What is the probability of rolling a sum of 7 with two dice?

A) \( \frac{1}{6} \)
B) \( \frac{1}{12} \)
C) \( \frac{1}{36} \)
D) \( \frac{5}{36} \)

45. What is the area of a circle with radius 7?

A) 154
B) 149
C) 154
D) 157

46. What is the volume of a cylinder with radius 3 and height 5?

A) 45
B) 60
C) 90
D) 75

47. If \( f(x) = 3x^2 - 5x + 2 \), what is \( f(2) \)?

A) 0
B) 0
C) 5
D) 6

48. What is the slope of the line \( y = 3x + 2 \)?

A) 3
B) 0
C) -1
D) 2

49. What is the range of the function \( f(x) = x^2 \) for real \( x \)?

A) \( (-\infty, \infty) \)
B) \( [0, \infty) \)
C) \( (-1, 1) \)
D) \( [1, \infty) \)

50. What is the inverse of the function \( f(x) = 2x + 3 \)?

A) \( f^{-1}(x) = \frac{x - 3}{2} \)
B) \( f^{-1}(x) = 2x - 3 \)
C) \( f^{-1}(x) = 3x + 2 \)
D) \( f^{-1}(x) = 2x + 3 \)

Result

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