Prove by mathematical induction:
3^4n+2 + 5^2n+1 is a multiple of 14

Pn: 34n+2+52n+1 is a multiple of 14Pn: 34n+2+52n+1=14k, kP1: 36+53=729+125=854=1461hence P1 is trueLet Pm be truePm:34m+2+52m+1=14k', k'Pm+1:34m+1+2+52m+1+1=14k'', k''L.H.S.34m+1+2+52m+1+1=34m+4+2+52m+1+1=3.4 34m+2+52m+1+1Using Pm34m+2=14k'-52m+1=3.4 14k'-52m+1+52m+1+2=81 14k'-52m+1+52. 52m+1=1481k'-81.52m+1+25. 52m+1=1481k'-56.52m+1=1481k'-4.52m+1=14k'', where k''=81k'-4.52m+1Hence Pm+1 is true, therefore, Pn is true

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