Question

$in \: a \: certain \: ap \: the \: 24th \: term \: is \: twice \: the \: 10th \: term \: prove \: that \: the \: 72 nd \: term \: is \: twice \: the \: 34th \: term \:$
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Answer 1

Given,

Without actual division, find out the quotient when the sum of numbers 167, 716 and 671 is divided by:

(a)37

(b) 111

By adding 167 + 716 + 671 = 1554

Now we know that by expanding the numbers we can write as

➠ 167 = 1 x 100 + 6 x 10 + 7

➠ 716 = 7 x 100 + 1 x 10 + 6

➠ 671 = 6 x 100 + 7 x 10 + 1

Now,

⟾ 167 + 716 + 671  

⟾ 1 x 100 + 6 x 10 + 7 + 7 x 100 + 1 x 10 + 6 + 6 x 100 + 7 x 10 + 1

⟾  (1 + 7 + 6)100 + (6 + 1 + 7)10 + (7 + 6 + 1) 14 x 100 + 14 x 10 + 14

⟾ 14(100 + 10 + 1)

⟾  14 x 111

⟾  14 x 3 x 37 (3 x 37 = 111)

⟾ 42 x 37

So quotient is 42 when 1554 is divided by 37.

Similarly for 111 we get

⟾  14 x 1 x 111

⟾  14 x 111  

So quotient is 14 when 1554 is divided by 111

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