Finding Common Ground

List all the factors of the following numbers: (a) 90 (b) 105 (c) 132 (d) 360 (this number has 24 factors) (e) 840 (this number has 32 factors)

After observing a few prime factorisations, Anshu claims "The larger a number is, the longer its prime factorisation will be". What do you think of Anshu's claim?

Find the common factors and the HCF of the following numbers: (a) 50, 60 (b) 140, 275 (c) 77, 725 (d) 370, 592

Find the HCF of the following numbers: (a) 24, 180 (b) 42, 75, 24 (c) 240, 378 (d) 400, 2500 (e) 300, 800

Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = $6 \times 12$ and 144 = $8 \times 18$. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?

Make a general statement about the HCF for the following pairs of numbers. You could consider examples before coming up with general statements. Look for possible explanations of why they hold. (a) Two consecutive even numbers (b) Two consecutive odd numbers (c) Two even numbers (d) Two consecutive numbers (e) Two co-prime numbers

The LCM of 3 and 24 is 24 (it is one of the two given numbers). (a) Find more such number pairs where the LCM is one of the two numbers. (b) Make a general statement about such numbers. Describe such number pairs using algebra.

Make a general statement about the LCM for the following pairs of numbers. You could consider examples before coming up with these general statements. Look for possible explanations of why they hold. (a) Two multiples of 3 (b) Two consecutive even numbers (c) Two consecutive numbers (d) Two co-prime numbers

(a) Is $5 \times 7$ × $11 \times 11$ a multiple of $5 \times 7$ × $7 \times 11$ × 2? (b) Is $5 \times 7$ × $11 \times 11$ a factor of $5 \times 7$ × $7 \times 11$ × 2?

Find the HCF and LCM of the following (state your answers in the form of prime factorisations): (a) $3 \times 3$ × $5 \times 7$ × 7 and $12 \times 7$ × 11 (b) 45 and 36

Find two numbers whose HCF is 1 and LCM is 66.

A cowherd took all his cows to graze in the fields. The cows came to a crossing with 3 gates. An equal number of cows passed through each gate. Later at another crossing with 5 gates again an equal number of cows passed through each gate. The same happened at the third crossing with 7 gates. If the cowherd had less than 200 cows, how many cows did he have?

The length, width, and height of a box are 12 cm, 18 cm, and 36 cm respectively. Which of the following sized cubes can be packed in this box without leaving gaps? (a) 9 cm (b) 6 cm (c) 4 cm (d) 3 cm (e) 2 cm

Among the numbers below, which is the largest number that perfectly divides both 306 and 36? (a) 36 (b) 612 (c) 18 (d) 3 (e) 2 (f) 360

Find the smallest number that is divisible by 3, 4, 5 and 7, but leaves a remainder of 10 when divided by 11.

A dog is chasing a rabbit that has a head start of 150 feet. It jumps 9 feet every time the rabbit jumps 7 feet. In how many leaps does the dog catch up with the rabbit?

What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 8, 9, 10? Do you remember the answer from Grade 6, Chapter 5?

Here is a problem posed by the ancient Indian Mathematician Mahaviracharya (850 C.E.). Add together 8/15, 1/20, 7/36, 11/63 and 1/21. What do you get? How can we find this sum efficiently?