MATHS 2 Marks Questions

LCM of two different prime numbers, m and n, is their product of m × n.
Since both m and n are prime numbers, their only common factor is 1.
Hence, LCM is always greater than both individual numbers.
Hence, option (c) is correct.

Let a = 5 × 7 × 11 × 11
6 = 5 × 7 × 7 × 11 × 2
For a to be a factor of b, b must contain all the prime factors of a, in equal or higher powers.
But here b has only one 11 (a has two 11s), so b does not include all factors of a.
Hence, 5 × 7 × 11 × 11 is not a factor of 5 × 7 × 7 × 11 × 2.

Here let a = 5 × 7 × 11 × 11 and b = 5 × 7 × 7 × 11 × 2
For a number a to be a multiple of number b, all prime factors of b must be present in a with at least the same power.
Number a does not have the prime factor 2.
Number a has 71, while number b has 7.
Hence, a is not a multiple of b.

Here 222, 370


222 = 2 × 3 × 37 and 370 = 2 × 5 × 37
∴ LCM(222, 370) = 2 × 3 × 5 × 37 = 1110.

Here 105, 195, and 65
Now 105 = 3 × 5 × 7
195 = 3 × 5 × 13
65 = 5 × 13
∴ LCM(105, 195, 65) = 3 × 5 × 7 × 13 = 1365

Here 36 = 2 × 2 × 3 × 3
(Two occurrences of 2s and two occurrences of 3s)
and 54 = 2 × 3×3×3
(One occurrence of 2, and three occurrences of 3s)
∴ LCM(36, 54) = 2 × 2 × 3 × 3 × 3 = 4 × 27 = 108
(One occurrence of 2, and three occurrences of 3s)
∴ LCM(36, 54) = 2 × 2 × 3 × 3 × 3 = 4 × 27 = 108

Here 30 = 2 × 3 × 5
(One occurrence of 2, one occurrence of 3, and one occurrence of 5)
and 72 = 2 × 2 × 2 × 3 × 3
(Three occurrences of 2s and two occurrences of 3s)
∴ LCM (30, 72) = 2 × 2 × 2 × 3 × 3 × 5
(Three occurrences of 2s and two occurrences of 3s, and one occurrence of 5)
= 8 × 9 × 5
= 360

Here 300 and 800

∴ HCF(300, 800) = 2 × 2 × 5 × 5 = 100.

Here 400 and 2500

∴ HCF(400, 2500) = 2 × 2 × 5 × 5 = 100.

Here 240 and 378

Hence, HCF(240, 378) = 2 × 3 = 6

Now

Clearly only common prime factor is 3.
∴ HCF (42, 75, 24) = 3.

Given 24, 180

Clearly, the common prime factors are 2 and 3.
∴ HCF (24, 180) = 2 × 2 × 3 = 4 × 3 = 12

840
Here


∴ 840 = 2 × 2 × 2 × 3 × 5 × 7
Hence, factors of 840 are 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840.
Total factors = 32

360
Here

∴ 360 = 2 × 2 × 2 × 3 × 3 × 5
Hence, factors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360.
Total factors = 24

132
Here

∴ 132 = 2 × 2 × 3 × 11
Hence, factors of 132 are 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132.
Total factors = 12

105
Here

∴ 105 = 3 × 5 × 7
Hence, factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105.
Total factors = 8

90
Here

∴ 90 = 2 × 3 × 3 × 5
Hence, factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
Total factors = 12