Math Olympiad Test: Real Numbers- 1 - Class 10 MCQ

L.C.M. of 144, 180, 192
144 = 2⁴ × 3²
180 = 2² × 3² × 5
192 = 2⁶ × 3
∴ L.C.M. = 2⁶ × 3² × 5 = 2880

Here 445 - 4 = 441; 572 - 5 = 567, 699 - 6 = 693
Now we have to find H.C.F. of 441, 567, 693 is
441 = 3 × 3 × 7 × 7
567 = 3 × 3 × 3 × 3 × 7
693 = 3 × 3 × 7 × 11
The common factors are 3 × 3 × 7 = 63.

Solve for the HCF of the given numbers by deducting the given remainders
Subtracting the remainders from numbers we get,
⇒ 398 − 7 = 391 [As the remainder is 7]
⇒ 436 − 11 = 425 [As the remainder is 11]
⇒ 542 − 15 = 527 [As the remainder is 15]
HCF of these new numbers is the largest number that divides 398, 436, 542 leaving respective remainders.
Factors of 391 = 17 × 23
Factors of 425 = 17 × 52
Factors of 527 = 17 × 31
The HCF of 391, 425, 527 is17.
Hence, the largest positive integer that will divide 398, 436 and 542, leaving remainders7, 11 and 15 respectively is 17.

In order to find the largest number which divides 615 and 963 leaving remainder 6 in each case, we need to deduct the remainder 6 from both the numbers and find their HCF.
615 − 6 = 609
963 − 6 = 957
Prime Factorising both 609 and 657 we get,
609 = 3 x 3 x 29  and
957 = 3 x 11 x 29
HCF of 609, 957 = 3 × 29
= 3 x 29 = 87
∴ The largest number which divides 615 and 963 leaving remainder 6 in each case is 87.

The smallest number which when divided by 35, 56 and 91 = LCM of 35, 56 and 91
35 = 5 x 7
56 = 2 x 2 x 2 x 7
91 = 7 x 13
LCM = 7 x 5 x 2 x 2 x 2 x 13 = 3640
The smallest number that when divided by 35, 56, 91 leaves a remainder 7 in each case = 3640 + 7 = 3647.

Given H.C.F. of 408 and 1032 = 24 and
1032m - 408 × 5 = 24
⇒ 1032m = 24 + 2040
⇒ 1032m = 2064
⇒ m = 2064/1032
= 2

Using the factor tree for prime factorization, we have:

We have 196 = 2² × 7²
2 + 2  = 4

5005 can be expressed as product of prime factor as:
5005 = 5 × 7 × 11 × 13
Thus, 5005 can be expressed as product of prime factors as 5005 = 5 × 7 × 11 × 13.

product of number = LCM x HCF
a x b = LCM x HCF
Other number

6ⁿ - 5ⁿ = 6¹ - 5¹ = 1
6² - 5² = 36 - 25 = 11 and so on