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Question: If \({9^7} - {7^9}\) is divisible by 2n, then find the greatest value of n ?...

If 97−79{9^7} - {7^9} is divisible by 2n, then find the greatest value of n ?

Explanation

Solution

Hint : We will solve this question using binomial expansion.
Binomial expansion – The binomial theorem is the method of expanding an expression which has been raised to any finite power.
Binomial expression – A binomial expression is an algebraic expression which contains two dissimilar terms. Ex. a+b,a3+b3a + b,{a^3} + {b^3} etc.
Binomial theorem :
(a+b)n=nC0anb0+nC1a(n−1)b1+nC2a(n−2)b2+.....+nCna0bn{(a + b)^n}{ = ^n}{C_0}{a^n}{b^0}{ + ^n}{C_1}{a^{(n - 1)}}{b^1}{ + ^n}{C_2}{a^{(n - 2)}}{b^2} + .....{ + ^n}{C_n}{a^0}{b^n}

Complete step by step solution :
97−79{9^7} - {7^9}
We can write it as, (8+1)7+(8−1)9{(8 + 1)^7} + {(8 - 1)^9}
or, (1+8)7+(−1)×(1−8)9{(1 + 8)^7} + ( - 1) \times {(1 - 8)^9}
(1+8)7−(1−8)9{(1 + 8)^7} - {(1 - 8)^9}
By using binomial expansion :
(1+x)n=nC0(1)nx0+nC1(1)(n−1)x1+.....+nCn(1)0xn{(1 + x)^n}{ = ^n}{C_0}{(1)^n}{x^0}{ + ^n}{C_1}{(1)^{(n - 1)}}{x^1} + .....{ + ^n}{C_n}{(1)^0}{x^n}
(1+8)7−(1−8)9{(1 + 8)^7} - {(1 - 8)^9}
=[(7C080+7C181+7C282+7C383+.....+7C787)]= \left[ {\left( {^7{C_0}{8^0}{ + ^7}{C_1}{8^1}{ + ^7}{C_2}{8^2}{ + ^7}{C_3}{8^3} + .....{ + ^7}{C_7}{8^7}} \right)} \right] −[(9C0(−8)0+9C1(−8)1+9C2(−8)2+9C3(−8)3+.....+9C9(−8)9)] - \left[ {\left( {^9{C_0}{{( - 8)}^0}{ + ^9}{C_1}{{( - 8)}^1}{ + ^9}{C_2}{{( - 8)}^2}{ + ^9}{C_3}{{( - 8)}^3} + .....{ + ^9}{C_9}{{( - 8)}^9}} \right)} \right]
=1+7C181+7C282+7C383+7C484+7C585+7C686+7C787= 1{ + ^7}{C_1}{8^1}{ + ^7}{C_2}{8^2}{ + ^7}{C_3}{8^3}{ + ^7}{C_4}{8^4}{ + ^7}{C_5}{8^5}{ + ^7}{C_6}{8^6}{ + ^7}{C_7}{8^7}
−1+9C181−9C282+9C383−9C484+9C585−9C686+9C787−9C888+9C989- 1{ + ^9}{C_1}{8^1}{ - ^9}{C_2}{8^2}{ + ^9}{C_3}{8^3}{ - ^9}{C_4}{8^4}{ + ^9}{C_5}{8^5}{ - ^9}{C_6}{8^6}{ + ^9}{C_7}{8^7}{ - ^9}{C_8}{8^8}{ + ^9}{C_9}{8^9}
=(7×8)+(9×8)+82[(7C2+7C381+7C482+.....)−(9C2−9C381+.....)]= (7 \times 8) + (9 \times 8) + {8^2}\left[ {\left( {^7{C_2}{ + ^7}{C_3}{8^1}{ + ^7}{C_4}{8^2} + .....} \right) - \left( {^9{C_2}{ - ^9}{C_3}{8^1} + .....} \right)} \right]
Term : [(7C2+7C381+7C482+.....)−(9C2−9C381+.....)]\left[ {\left( {^7{C_2}{ + ^7}{C_3}{8^1}{ + ^7}{C_4}{8^2} + .....} \right) - \left( {^9{C_2}{ - ^9}{C_3}{8^1} + .....} \right)} \right] is constant.
So, let this term be k
Then, we have
=(7×8)+(9×8)+82k= (7 \times 8) + (9 \times 8) + {8^2}k
=56+72+82k= 56 + 72 + {8^2}k
=132+82k= 132 + {8^2}k
=82(4+k)= {8^2}(4 + k)
So,
97−79=82(4+k){9^7} - {7^9} = {8^2}(4 + k)
=2×32(4+k)= 2 \times 32(4 + k)
As, we have to prove 97−79{9^7} - {7^9} as a multiple of 2.
So, we write 82=64{8^2} = 64 as 2×322 \times 32.
And also, 32(4+k)32(4 + k) is another constant.
Therefore, 97−79=2(32c)=2n{9^7} - {7^9} = 2(32c) = 2n
Where n=32n = 32

Note : Make sure you do not apply the formula/concept of permutation here .There is a high chance of students getting confused whether to apply permutation or combination in these kinds of problems.