MATHS 5 Marks Questions

Interpretation is that when a number k is called, the children are grouped into rows of size k, and “no one got out” means the children formed complete rows with no remainder.
So: “No one got out” when 6 was called ⇒ the total number N is divisible by 6.
“No one got out” when 9 was called ⇒ N is divisible by 9.
“Some people got out” when 10 was called ⇒ N is not divisible by 10.
If N is divisible by both 6 and 9, then it must be divisible by their LCM:
LCM (6, 9) = 18.
So N is a multiple of 18, but not a multiple of 10.
Now check the options:
(a) 72 = 18 × 4 – divisible by 18 and not by 10 → possible.
(b) 90 = 18 × 5 – divisible by 18 but is divisible by 10 → not possible.
(c) 45 – not divisible by 18 → not possible.
(d) 3 – not divisible by 18 → not possible.
(e) 36 = 18 × 2 – divisible by 18 and not by 10 → possible.
So the numbers that could have been playing are 36 and 72.

Here, dimensions of box: Length = 12 cm, Width = 18 cm, Height = 36 cm
The size of the largest cube that can exactly fit (without gaps)
That means the side of the cube must exactly divide all three dimensions of the box.
So, we need to find the HCF (Highest Common Factor) of 12, 18, and 36.
Prime factorisation 12 = 2 × 2 × 3, 18 = 2 × 3 × 3, and 36 = 2 × 2 × 3 × 3
Common factors = 2 × 3 = 6
HCF = 6 cm
The cube must have a side length equal to a factor of the HCF.
From the options:
(a) 9 cm ✗ (9 doesn’t divide 12 evenly)
(b) 6 cm ✓ (divides 12, 18, and 36 exactly)
(c) 4 cm ✗ (doesn’t divide 18 evenly)
(d) 3 cm ✓ (also divides all)
(e) 2 cm ✓ (also divides all)
Hence, the largest possible cube that fits without gaps is (b) 6 cm.

Here are more number pairs where the LCM of the two numbers is one of the given numbers:
(i) LCM of 2 and 4: The LCM is 4.
Prime factorization of 2: 2
Prime factorization of 4: 2 × 2
LCM (2, 4) = 2 × 2 = 4
(ii) LCM of 5 and 10: The LCM is 10.
Prime factorization of 5: 5
Prime factorization of 10: 2 × 5
LCM (5, 10) = 2 × 5 = 10
(iii) LCM of 6 and 12: The LCM is 12.
Prime factorization of 6: 2 × 3
Prime factorization of 12: 2 × 2 × 3
LCM (6, 12) = 2 × 2 × 3 = 12
(iv) LCM of 7 and 49: The LCM is 49.
Prime factorization of 7: 7
Prime factorization of 49: 7 × 7
LCM (7, 49) = 49
(v) LCM of 10 and 100: The LCM is 100.
Prime factorization of 10: 2 × 5
Prime factorization of 100: 2 × 2 × 5 × 5
LCM (10, 100) = 2 × 2 × 5 × 5 = 100