PSLEThere are two numbers. Both are 2-digit numbers. One is 2 times the other. They have exactly five common factors. Three of the common factors are 1, 2 and 4.
- What are the other 2 common factors? Give your answers in ascending order. (Eg 1, 2)
- What are the two numbers? Give your answers in ascending order. (Eg 1, 2)
(a)
To have common factors 1, 2 and 4, the two numbers must be divisible by 1, 2 and 4.
Lowest common multiple of 2 and 4: 4
4
= 1 x 4
= 2 x 2
Factors of 4: 1, 2 and 4
However, 4 cannot be the smaller number because it has only 3 factors and is not a 2-digit number.
Multiples of 4: 4, 8, 12, 16
The smaller number must have exactly five common factors. So, it must be a square number.
16
= 1 x 16
= 2 x 8
= 4 x 4
Factors of 16: 1, 2, 4, 8, 16
So, 16 has 5 factors and is divisible by 1, 2 and 4, it is the smaller number.
The other two common factors are 8 and 16.
(b)
The smaller number, 16, is the highest common factor.
The larger number is 2 times of the other.
Larger number
= 2 x 16
= 32
The two numbers are 16 and 32.
Answer(s): (a) 8, 16; (b) 16, 32