JEE MAIN - Mathematics (2021 - 25th February Evening Shift - No. 19)
The total number of two digit numbers 'n', such that 3n + 7n is a multiple of 10, is __________.
Answer
45
Explanation
$$ \because $$ $${7^n} = {(10 - 3)^n} = 10k + {( - 3)^n}$$
$${7^n} + {3^n} = 10k + {( - 3)^n} + {3^n}$$
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$$ \therefore $$ 3n = 32t = (10 $$-$$ 1)t
= 10p + ($$-$$1)t
= 10p $$\pm$$ 1
$$ \therefore $$ if n = even then 7n + 3n will not be multiply of 10
So if n is odd then only 7n + 3n will be multiply of 10
$$ \therefore $$ n = 11, 13, 15, ..........., 99
$$ \therefore $$ Ans : 45
$${7^n} + {3^n} = 10k + {( - 3)^n} + {3^n}$$
$$ \therefore $$ 3n = 32t = (10 $$-$$ 1)t
= 10p + ($$-$$1)t
= 10p $$\pm$$ 1
$$ \therefore $$ if n = even then 7n + 3n will not be multiply of 10
So if n is odd then only 7n + 3n will be multiply of 10
$$ \therefore $$ n = 11, 13, 15, ..........., 99
$$ \therefore $$ Ans : 45
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