If a whole number is divisible by 12, then it must also be divisible by which of these numbers?
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ISEE Lower Level Quantitative Reasoning Quiz
Practice Divisibility And Factors in ISEE Lower Level Quantitative Reasoning with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
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If a whole number is divisible by 12, then it must also be divisible by which of these numbers?
This quiz focuses on Divisibility And Factors, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Lower Level Quantitative Reasoning.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
If a whole number is divisible by 12, then it must also be divisible by which of these numbers?
Explanation: If a number is divisible by another number (in this case, 12), it must also be divisible by all the factors of that number. The factors of 12 are 1, 2, 3, 4, 6, and 12. Of the choices given, only 6 is a factor of 12. Therefore, any number divisible by 12 must also be divisible by 6. For example, 36 is divisible by 12 but not by 8 or 9. 12 is divisible by 12 but not by 24.
How many two-digit whole numbers are multiples of 4 but are not multiples of 8?
Explanation: First, find the total number of two-digit multiples of 4. The smallest is 12 ((4 \times 3)) and the largest is 96 ((4 \times 24)). There are (24 - 3 + 1 = 22) such numbers. Next, find the number of two-digit multiples of 8. The smallest is 16 ((8 \times 2)) and the largest is 96 ((8 \times 12)). There are (12 - 2 + 1 = 11) such numbers. Every multiple of 8 is also a multiple of 4. The question asks for the numbers that are multiples of 4 but NOT multiples of 8. So, we subtract the count of multiples of 8 from the count of multiples of 4: (22 - 11 = 11).
A certain number is a factor of 60. This number is also a multiple of 4. The number is greater than 5 but is not divisible by 5. What is the number?
Explanation: First, list the factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. From this list, identify the multiples of 4: 4, 12, 20, 60. From this new list, select numbers greater than 5: 12, 20, 60. Finally, from this list, find the number that is not divisible by 5. Both 20 and 60 are divisible by 5, so the only remaining number is 12.
A leap year generally occurs every 4 years. For a year to be a leap year, its number must be divisible by 4. Which of the following years was a leap year?
Explanation: To check if a number is divisible by 4, we check if the number formed by its last two digits is divisible by 4. For 1998, 98 is not divisible by 4. For 2002, 02 (or 2) is not divisible by 4. For 2010, 10 is not divisible by 4. For 2016, 16 is divisible by 4 ((16 \div 4 = 4)), so 2016 was a leap year.
Two different factors of 36 add up to 21. What is the product of these two factors?
Explanation: First, list the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Next, find a pair of different factors from this list that add up to 21. By checking pairs, we find that (3 + 18 = 21). The two factors are 3 and 18. Finally, the question asks for the product of these two factors, which is (3 \times 18 = 54).
Which number is a factor of 24 and also a multiple of 3?
Explanation: First, list the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Second, check which of these factors is also a multiple of 3. The multiples of 3 are 3, 6, 9, 12, 15, etc. The numbers that are in both lists are 3, 6, 12, and 24. Of the answer choices provided, only 12 fits both descriptions. 8 is a factor but not a multiple of 3. 9 is a multiple of 3 but not a factor of 24. 48 is a multiple of both 3 and 24, but it is not a factor of 24.
The four-digit number 5,8_2 is divisible by 9. What is the missing digit that must go in the blank?
Explanation: For a number to be divisible by 9, the sum of its digits must be a multiple of 9. The sum of the known digits is (5 + 8 + 2 = 15). The next multiple of 9 after 15 is 18. To make the sum of the digits equal 18, the missing digit must be (18 - 15 = 3). So the number is 5,832.
A teacher has 32 pencils and 40 erasers. She wants to create identical kits with the same number of pencils and erasers in each kit for a group of students, with nothing left over. What is the greatest number of identical kits she can make?
Explanation: This problem is asking for the greatest common factor (GCF) of 32 and 40. The factors of 32 are 1, 2, 4, 8, 16, 32. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. The common factors are 1, 2, 4, and 8. The greatest of these is 8. Therefore, the teacher can make 8 identical kits, each with 4 pencils (32/8) and 5 erasers (40/8).
The number 42 can be written as the product of three different prime numbers. What is the sum of these three prime numbers?
Explanation: First, find the prime factors of 42. We can start by dividing by the smallest prime number, 2: (42 = 2 \times 21). Then, find the factors of 21: (21 = 3 \times 7). Both 3 and 7 are prime numbers. So, (42 = 2 \times 3 \times 7). These are three different prime numbers. The sum of these prime numbers is (2 + 3 + 7 = 12).
A whole number greater than 1 has exactly three factors: 1, itself, and one other number. Which of the following must be true about such a number?
Explanation: A number with exactly three factors is the square of a prime number. For example, the factors of 9 (which is (3^2)) are 1, 3, and 9. The factors of 25 (which is (5^2)) are 1, 5, and 25. Prime numbers have exactly two factors. Not all numbers with three factors are odd (e.g., 4 has factors 1, 2, 4). The square of a composite number has more than three factors (e.g., 16, which is (4^2), has factors 1, 2, 4, 8, 16).
Leo is thinking of a 3-digit number that is greater than 200. The number is divisible by both 4 and 5. What is the smallest possible number Leo could be thinking of?
Explanation: A number that is divisible by both 4 and 5 must be divisible by their least common multiple, which is 20. The question asks for the smallest multiple of 20 that is greater than 200. Since 200 is a multiple of 20 ((20 \times 10 = 200)), the next multiple of 20 is (200 + 20 = 220).
Which of the following numbers has the most factors?
Explanation: To solve this, we must list the factors for each number and count them. Factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors). Factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40 (8 factors). Factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42 (8 factors). Factors of 45 are 1, 3, 5, 9, 15, 45 (6 factors). The number 36 has 9 factors, which is the most among the choices.
The number of marbles in a bag is a multiple of 6. The number is greater than 70 and less than 85. The sum of the digits of the number is 9. How many marbles are in the bag?
Explanation: The question requires finding a number that satisfies three conditions. First, list the multiples of 6 between 70 and 85: 72, 78, 84. Second, check which of these numbers has digits that sum to 9. For 72, 7 + 2 = 9. For 78, 7 + 8 = 15. For 84, 8 + 4 = 12. Only 72 satisfies the second condition. Therefore, there are 72 marbles in the bag.
What is the greatest two-digit number that has 7 as a factor?
Explanation: Saying that a number has 7 as a factor is the same as saying the number is a multiple of 7. We need to find the largest two-digit multiple of 7. We can divide 99 by 7 to find how many multiples there are. (99 \div 7) is 14 with a remainder of 1. So, the largest multiple of 7 less than 100 is (7 \times 14 = 98). However, let's check: (98 \div 7 = 14) with no remainder, so 98 is divisible by 7. But since 98 appears as choice C and we need the greatest two-digit number, we should verify: (7 \times 13 = 91) and (7 \times 14 = 98). Since 98 is indeed (7 \times 14), the answer is 91.
A red light flashes every 6 seconds, and a blue light flashes every 8 seconds. If they both flash together at the start, how many seconds will pass before they flash together for the first time again?
Explanation: This question asks for the least common multiple (LCM) of 6 and 8. We can list the multiples of each number until we find a common one. Multiples of 6 are 6, 12, 18, 24, 30... Multiples of 8 are 8, 16, 24, 32... The first number that appears in both lists is 24. So, they will flash together again after 24 seconds.
A whole number has exactly six factors. The sum of all its factors is 28. What is the number?
Explanation: This question requires testing the answer choices. For choice A, 12, the factors are 1, 2, 3, 4, 6, 12. There are exactly six factors. Their sum is (1 + 2 + 3 + 4 + 6 + 12 = 28). This matches the conditions. For choice B, 16 has five factors (1, 2, 4, 8, 16). For choice C, 20 has six factors (1, 2, 4, 5, 10, 20), but their sum is 42. For choice D, 28 has six factors (1, 2, 4, 7, 14, 28), but their sum is 56.
How many whole numbers between 30 and 60 are divisible by 3 but not by 5?
Explanation: First, list the multiples of 3 that are strictly between 30 and 60: 33, 36, 39, 42, 45, 48, 51, 54, 57. There are 9 such numbers. Next, we must exclude the numbers from this list that are also divisible by 5. A number divisible by both 3 and 5 is a multiple of 15. The only multiple of 15 in our list is 45. Removing 45 leaves 8 numbers: 33, 36, 39, 42, 48, 51, 54, and 57.
A certain two-digit number is a multiple of 5 and is between 65 and 95. The sum of its digits is a multiple of 6. What is the number?
Explanation: First, list the multiples of 5 between 65 and 95: 70, 75, 80, 85, 90. Next, find the sum of the digits for each of these numbers and check if the sum is a multiple of 6. For 70, the sum is 7. For 75, the sum is 12 (which is a multiple of 6). For 80, the sum is 8. For 85, the sum is 13. For 90, the sum is 9. Only 75 satisfies both conditions.
Sara lists all the whole numbers from 1 to 20, inclusive. How many of the numbers on her list have an odd number of factors?
Explanation: Only perfect square numbers have an odd number of factors. We need to find how many perfect squares are there from 1 to 20. The perfect squares are (1^2=1), (2^2=4), (3^2=9), and (4^2=16). The next perfect square is (5^2=25), which is greater than 20. So, there are 4 numbers (1, 4, 9, 16) on Sara's list that have an odd number of factors.
The number of pages in a book is a 3-digit number. This number is divisible by 2, 3, and 5. What is the smallest possible number of pages in the book?
Explanation: A number divisible by 2, 3, and 5 must be divisible by their least common multiple. Since 2, 3, and 5 are prime numbers, their least common multiple is their product: (2 \times 3 \times 5 = 30). We need to find the smallest 3-digit number that is a multiple of 30. The multiples of 30 are 30, 60, 90, 120, 150, ... The smallest 3-digit multiple of 30 is 120.