Is the following statement true, partially true, or false: '5 is a factor of 35, and 35 is a multiple of 5'?
AFalse — a number can only be one or the other, not both
BPartially true — 35 is a multiple of 5, but 5 cannot be a factor because it is smaller than 35
CTrue — both statements describe the same multiplication relationship from different directions
DPartially true — 5 is a factor of 35, but 35 is too large to be a multiple of 5
Both statements are simultaneously and completely true. Because 5 × 7 = 35, we know that 5 is a factor of 35 (it divides evenly into 35) AND that 35 is a multiple of 5 (it results from multiplying 5 by a whole number). Factors and multiples are two sides of the same multiplication relationship, not competing labels.
Question 2 Multiple Choice
A student claims that the factors of 15 include the number 30 because 30 is related to 15 by multiplication. What is wrong with this reasoning?
ANothing — 30 is indeed a factor of 15
BFactors of a number must be less than or equal to the number, and 30 is larger than 15
C30 is only a factor if it divides into 15 with no remainder, and 30 × 0.5 = 15 counts
DFactors only include odd numbers
A factor of a number must divide into it evenly — and a factor is always less than or equal to the original number. 30 does not divide into 15 evenly (15 ÷ 30 = 0.5, which is not a whole number), so 30 is not a factor of 15. 30 is actually a multiple of 15. The common misconception is thinking factors can be larger than the number they divide; they cannot.
Question 3 True / False
Since factors come from multiplication, the factors of a number can sometimes be larger than the number itself.
TTrue
FFalse
Answer: False
Factors must divide into the number evenly, which means every factor is less than or equal to the number. A factor of n is a whole number that divides n with no remainder — this automatically means it cannot exceed n. The only way a × b = n with a > n would require b to be less than 1, which is not a whole number. Multiples, not factors, grow larger than the original number.
Question 4 True / False
The number 6 is simultaneously a factor of 6 and the smallest positive multiple of 6.
TTrue
FFalse
Answer: True
Every number is a factor of itself (n × 1 = n, so n divides n evenly) and is also its own smallest positive multiple (n × 1 = n). The number sits at the top of its own factor list and at the bottom of its own multiple list. This is not a special case — it applies to every positive whole number.
Question 5 Short Answer
A classmate says '12 is a factor of 3 because 3 goes into 12.' What is the error, and what is the correct relationship between 3 and 12?
Think about your answer, then reveal below.
Model answer: The error is a common confusion of direction. '3 goes into 12' means 3 divides 12 evenly — so 3 is a factor of 12, not the other way around. 12 is a multiple of 3 (because 3 × 4 = 12). The classmate has the labels reversed: the smaller number (3) is the factor, and the larger number (12) is the multiple.
A reliable memory tool: factors are smaller (or equal), multiples are larger (or equal). Because 3 × 4 = 12, we say '3 is a factor of 12' and '12 is a multiple of 3.' The phrase '3 goes into 12' is a division way of expressing the same fact — and the number doing the dividing (3) is the factor.