xx is a two digit integer, the first term of a sequence of consecutive

Question

xx is a two digit integer, the first term of a sequence of consecutive odd positive integers. If the sum of xx and the last term equals 100, and the product of xx and the last term is a number whose four digits equal, in order, x-1, x-1, x-2, x-2, then what is the sum of the four digits?

A. 6
B. 10
C. 14
D. 18
E. 22

A. 6
B. 10
C. 14
D. 18
E. 22

chetan2u · Answer

Best is to use options.

So sum = x-1 + x-1 + x-2 + x-2 = 4x-6
As xx is odd, x has to be odd.

A. 4x-6 = 6 ……….x=3
Thus, the number is 33, and the last term is 100-33 = 67.
33*67 = 2211…YES

Let us test others too.

B. 4x-6 = 10 ……….x=4.. DISCARD as x is not odd. But we can still check, if say question did not mention odd. 
Thus, the number is 44, and the last term is 100-44 = 56.
44*56: NO as the units digit will be 4, while it should be x-2 or 2

C. 4x-6 = 14 ……….x=5
Thus, the number is 55, and the last term is 100-55 = 45.
55*45: NO as the units digit will be 5, while it should be x-2 or 3

D. 4x-6 = 18 ……….x=6 .. DISCARD as x is not odd. But we can still check, if say question did not mention odd. 
Thus, the number is 66, and the last term is 100-66 = 34.
66*34: the units digit will be 4, which would be ok, but 66*34 will never be 5abc or 5544.

B. 4x-6 = 22 ……….x=7
Thus, the number is 77, and the last term is 100-77 = 23.
77*23: NO as the units digit will be 1, while it should be x-2 or 5

A

sanjitscorps18 · Answer

xx could be 11, 33, If 55 or 77 or 99 but we cannot have a number larger than xx that could be used to create a sum of 100. For e.g. 77 requires 23 to get 100 but 23 cannot be a term of a sequence of consecutive positive odd numbers starting 77.

Hence only possibilities are 11 or 33

if x = 1 then xx = 11 and that would require 89 but when multiplied it gives a 3 digit number, hence we ignore that

if x = 3 then xx = 33, hence last term would be 67. Now 33 multiplied with 67 gives 2211 that is exactly of the type x-1,x-1,x-2,x-2 when x = 3

Hence adding all of them gives 6.

Option A

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xx is a two digit integer, the first term of a sequence of consecutive

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xx is a two digit integer, the first term of a sequence of consecutive

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