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Real Numbers
Check whether 6n can end with the digit 0 for any natural number n
If the number 6n, for any natural number n, ends with digit 0, then it is divisible by 5. That is, the prime factorisation of 6n contains the prime 5. This is not possible because the primes in the factorisation of 6n are 2 and 3 and the uniqueness of the Fundamental Theorem of Arithmetic guarantees that there are no other primes in the factorisation of 6n.
So, there is no natural number n for which 6n ends with digit zero.
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Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
Express each number as a product of its prime factors: (i) 140
Express each number as a product of its prime factors: (ii) 156
Express each number as a product of its prime factors: (ii) 156 (iii) 3825 (iv) 5005 (v) 7429
Express each number as a product of its prime factors: (iv) 5005
Express each number as a product of its prime factors: (v) 7429
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (i) 26 and 91
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (ii) 510 and 92
Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers. (iii) 336 and 54
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