Factors & Multiples: Level 1
1.
What is the remainder when 100 is divided by 9?
Solution (A): \(100 \div 9\). We know that \(9 \times 11 = 99\).
\(100 – 99 = 1\).
The remainder is 1.
2.
What is the Least Common Multiple (LCM) of 4 and 10?
Solution (C): Multiples of 4 are {4, 8, 12, 16, 20, 24…}.
Multiples of 10 are {10, 20, 30…}.
The smallest number on both lists is 20.
3.
What is the Greatest Common Factor (GCF) of 16 and 24?
Solution (D):
Factors of 16 are {1, 2, 4, 8, 16}.
Factors of 24 are {1, 2, 3, 4, 6, 8, 12, 24}.
The largest number on both lists is 8.
4.
When \(x\) is divided by 5, the remainder is 2. What is the remainder when \(x+4\) is divided by 5?
Solution (E): We can add the remainders.
The remainder of \(x\) is 2. The remainder of 4 is 4.
\(2 + 4 = 6\).
Now, find the remainder of 6 when divided by 5.
\(6 \div 5 = 1\) with a remainder of 1.
5.
When \(x\) is divided by \(y\), the remainder is 6. Which of the following is a possible value for \(y\)?
Solution (C): The remainder must always be smaller than the divisor.
Since the remainder is 6, the divisor \(y\) must be greater than 6.
Of the options, only 7 is greater than 6.
6.
How many distinct positive factors does 12 have?
Solution (D): The factors of 12 are {1, 2, 3, 4, 6, 12}.
Counting them, we find there are 6 factors.
7.
If a positive integer \(n\) is divisible by 3 and 5, then \(n\) must be divisible by which of the following?
Solution (A): If \(n\) is divisible by 3 and 5, it must be a multiple of their Least Common Multiple (LCM).
Since 3 and 5 are prime, their LCM is \(3 \times 5 = 15\).
Therefore, \(n\) must be divisible by 15.
8.
Which of the following numbers has a remainder of 2 when divided by 6?
Solution (B): We test each option.
(A) \(12 \div 6 = 2\) rem 0.
(B) \(14 \div 6 = 2\) rem 2. (This is the answer).
(C) \(16 \div 6 = 2\) rem 4.
(D) \(18 \div 6 = 3\) rem 0.
(E) \(20 \div 6 = 3\) rem 2.
*Correction:* Both B and E have a remainder of 2. This is a flawed question. I will fix option E.
New (E) 22. \(22 \div 6 = 3\) rem 4.
8.
Which of the following numbers has a remainder of 2 when divided by 6?
Solution (B): We test each option.
(A) \(12 \div 6 = 2\) rem 0.
(B) \(14 \div 6 = 2\) rem 2. (This is the answer).
(C) \(16 \div 6 = 2\) rem 4.
(D) \(18 \div 6 = 3\) rem 0.
(E) \(22 \div 6 = 3\) rem 4.
9.
When the positive integer \(x\) is divided by 9, the remainder is 2. What is the remainder when \(3x\) is divided by 9?
Solution (D): We can multiply the remainder.
\(x\) has a remainder of 2.
\(3x\) has a remainder of \(3 \times 2 = 6\).
Since 6 is less than 9, the remainder is 6.
10.
What is the lowest positive integer that is divisible by 2, 3, and 4?
Solution (C): We need to find the Least Common Multiple (LCM) of {2, 3, 4}.
Prime factors:
2 = 2
3 = 3
4 = \(2^2\)
The LCM must have the highest power of each prime: \(2^2 \times 3^1 = 4 \times 3 = 12\).
Score: 0 / 10