Show that x is rational if: x2 = 0.0004
= 2 / 100 = 0.02, which is rational
Without doing any actual division, find which of the following rational numbers have terminating decimal representation:
(i) 7 / 16
(ii) 23 / 125
(iii) 9 / 14
(iv) 32 / 45
(v) 43 / 50
(vi) 17 / 40
(vii) 61 / 75
(viii) 123 / 250
(i) 7 / 16 = 7 / 2 x 2 x 2 x 2 denominator multiple of 2m x 5n hence terminating.
(ii) 23 / 125 = 23 / 5 x 5 x 5 denominator multiple of 2m x 5n hence terminating.
(iii) 9 / 14 = 9 / 7 x 2 denominator is not multiple of 2m x 5n hence non-terminating.
(iv) 32 / 45 = 32 / 3 x 3 x 5 denominator is not multiple of 2m x 5n hence non-terminating.
(v) 43 / 50 = 43 / 5 x 5 x 2 denominator multiple of 2m x 5n hence terminating.
(vi) 17 / 40 = 17 / 2 x 2 x 2 x 5 denominator multiple of 2m x 5n hence terminating.
(vii) 61 / 75 = denominator is not multiple of 2m x 5n hence non-terminating.
(viii) 123 / 250 = 123 / 5 x 5 x 5 x 2 denominator multiple of 2m x 5n hence terminating.
(i), (ii), (v), (vi) and (viii) will have terminating decimal.
Prove that 3 + √5 is an irrational number.
Find the greatest number that will divide 93, 111 and 129, leaving remainder 3 in each case.
93 - 3 = 90
111 - 3 = 108
129 - 3 = 126
Required number will be H.C.F of 90,108 and 126
H.C.F of 90 and 108
∴ The greatest number is 18.
If the product of two numbers is 540 and their HCF is 30, find their LCM.
HCF x LCM = 540
30 x LCM = 540
LCM = 540 / 30 = 18
π is an irrational number.
The sum of two irrational numbers is an irrational number.
If the product of two co-primes is 553, then their LCM = _____
∴ Product of the two numbers = HCF × LCM
Thus LCM = 553
The value of x in following factor tree is ____
2 × 21 = 42
3 × 14 = 42
6 × 7 = 42
Therefore, the factors of 42 in pairs are (1, 42), (2, 21), (3, 14) and (6, 7).
Product of any irrational number with a rational number 'x' is always rational. Then, x is ____.
The prime factorisation of 320 is____.
320 = 26 x 51
Any number ending with '0' must have ___ and ___ as its prime factors.
The factors of 10 = 1, 2, 5 and
The prime factors of 10 = 2 and 5
The factors of 20 = 1, 2, 4, 5, 10 and 20
The prime factors of 20 = 2 and 5
∴ Any number ending with '0' must have '2' and '5' as its prime factors.
HCF of two numbers is always a factor of their LCM.
Because HCF is a factor of both the numbers which are factors of their LCM.
Hence, HCF of two numbers is always a factor of their LCM.
Every even integer is of the form 2m, where m is an integer.
For Example : Let the value of m be -2, 4 & 0
Then the values for 2m = 2 × -2 , 2 × 4 & 2 × 0
= - 4 , 8 & 0
If p and q are relatively prime numbers, what is their LCM?
p = 1 × p
q = 1 × q
HCF of a and b = 1.
Their LCM = 1 × p × q
LCM of p and q = pq
For what value of n, 2n x 5n ends in 5.
2n × 5n = (2 × 5)n = (10)n
which always ends in zero, as 102 = 100, 103 = 1000,.....
Thus, there is no value of n for which (2n × 5n ) ends in 5.
LCM of two prime numbers is 1.
LCM of two or more ____ is equal to their product.
If a and b are two odd prime numbers, then a2 - b2 is ____(prime/composite/neither prime nor composite).
HCF of 870 and 225 is:
Factors of 225 = 5 × 5 × 3 × 3
Common factors = 5 × 3 = 15
H.C.F = 15
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